Revision history for Perl module Math::Prime::Util

0.75 2026-?

    - Requires Perl 5.8.1 or later.  Dropping Perl 5.6 support has been
      considered for years.  Supporting very old Perl builds with broken
      64-bit integer handling was increasingly time-consuming, especially
      as modern compiler and testing infrastructure no longer supports
      such old Perl versions.

    [API Changes]

    - prime_set_config does not allow changing the xs option.  Set MPU_NO_XS=1
      before loading the module to select the Pure Perl implementation.

    - When a function returns an integer result larger than native type size,
      it will use the user configured bigint class (default Math::BigInt).
      We no longer try to preserve the input bigint class.

    - random_bytes and entropy_bytes accept inputs 0 to 2^31-2 (max portable
      size for a Perl string).  The previously documented values were
      ambiguous (unsigned).  This makes them concrete.

    - urandomb and functions that produce n-bit random primes or semiprimes
      are limited to 2^32-1 bits.

    - irand64 always returns a random integer in [0 .. 2^64-1].  On 32-bit this
      means it will likely return a bigint.

    - random_ndigit_prime is limited to 2^32-1 digits.

    - jordan_totient(k,n) and powersum(n,k) now require exponent k to fit
      in an unsigned native integer.

    - factorial, subfactorial, bell_number, fubini, primorial, pn_primorial,
      catalan_number, partitions, partitionsq, and consecutive_integer_lcm
      will reject inputs that exceed implementation limits (input limited to
      unsigned native for XS and GMP, signed native for PP).

    - is_power($n,\$root) is now the preferred form for returning the maximal
      exponent and root.  No need for a dummy second argument.

    [ADDED]

    - is_safe_prime(n)                returns 1 if odd pos n and (n-1)/2 prime
    - fibonacci(k)                    Fibonacci number, same as lucasu(1,-1,k)
    - lucas_number(k)                 Lucas number, same as lucasv(1,-1,k)
    - aliquot_sum                     sum of proper divisors
    - prime_signature(n)              reverse sorted list of factor exponents
    - sopfr(n)                        sum of prime factors
    - sopf(n)                         sum of distinct prime factors
    - digital_root(n[,base])          iterated sum of digits (default base 10)
    - mult_digital_root(n[,base])     iterated product of digits
    - is_palindrome(n[,base])         is n a palindrome (default base 10)
    - is_harshad(n[,base])            is n a Harshad (Niven) number in base b
    - abundance(n)                    sigma(n)-2n => deficient/abundant/perfect
    - bell_number(n)                  Bell number, count of set partitions
    - catalan_number(n)               Catalan number
    - integer_complexity(n)           minimum 1s to represent n using + and *
    - multifactorial(n,k)             k-step multifactorial of n
    - dedekind_psi(n)                 Dedekind psi function, n*prod(1+1/p)
    - partitionsq(n)                  number of partitions into distinct parts
    - convergents(@A)                 list of convergents (p,q) for cfrac list
    - bestrational(x,dbound)          best rational (p,q) for x with q<=dbound
    - muladdint(n,m,a)                signed integer n*m + a
    - mulsubint(n,m,a)                signed integer n*m - a
    - toint(n)                        truncate input to integer (best form)
    - urandomr(lo,hi)                 random integer in [lo,hi] inclusive
    - vecprefixsum(@list)             prefix sum / cumulative sum of list
    - vecwindow { } $step,$size,@list calls block with windows of size objects
    - vecpairwise { } \@A,\@B         calls block for each pair from arrays
    - vecrsort(@L)                    numerically reverse sort integer list
    - vecrsorti(\@L)                  in-place numeric reverse sort a list ref
    - remove_factors(n,k)             returns r: n with all factors of k removed
    - remove_factors_exp(n,k)         as above, returns (r,e) e = times removed
    - floor_sum(n,m,a,b)              sum floor((a*i+b)/m) for i=0..n-1
    - inverse_sigma0(k,[lo,]hi)       array ref of n where sigma0(n) = k
    - inverse_sigma0_count(k,[lo,]hi) count of n where sigma0(n) = k
    - crootint(n,k)                   ceiling integer k-th root
    - reverse_digits(n[,base])        reverse digits of |n| in base b
    - irand32()                       random 32-bit integer (like irand)

    [FIXES]

    - drand now always returns below 1 before scaling in both XS and PP.

    - PP drand, srand, and csrand reject extra arguments, and csrand returns
      no value in both XS and PP.

    - PP uses timer and scheduler jitter if no normal entropy source is
      available, rather than deterministically seeding the CSPRNG with zero.

    - PP threads independently reseed their CSPRNG after interpreter cloning.

    - LambertW consistently handles floating-point rounding just below -1/e.

    - Fix rare nth_prime hangs and native overflow near the 64-bit limit.

    - XS setbinop and vec callbacks preserve lexical return values.  Sparse
      arrays passed to set functions are rejected safely.

    - Do not send bigint Lucas parameters to GMP versions before 0.53.

    - chebyshev_theta and chebyshev_psi had tables corrected for 3 ranges.

    - Fix overflow handling in Farey functions on 32-bit builds.

    - Fix native IV_MIN edge cases in several integer functions, including
      divrem.  Fix tdivrem with negative inputs on some Windows builds.

    - signint correctly handles positive bigint strings with leading zeroes.

    - trial_factor always removes factors 2, 3, and 5 with bigint inputs,
      including when given a smaller factor limit.  Match existing docs.

    - lastfor handles die/croak inside loops much better.

    - random_factored_integer dropped the array return for bigints.

    - srand synchronizes its complete, endian-independent UV seed with the
      GMP backend, including when called without an argument.

    - urandomm(n) accepted n=0 even though documented as positive.  Now rejects.

    - all divisors for large n are properly converted to bigint if needed.

    - Better input validation for many PP and XS paths, especially for
      random funcs, digit/base conversion, combinatorial counts, iterator
      arguments, and large size limits.

    - PP/XS/GMP consistency fixes.  Better returning of canonical bigint class.

    - miller_rabin_random requires nbases > 0, and no longer accepts a third
      argument.  Neither of these were documented.

    - prime_count_approx without GMP preserves more high-precision digits for
      very large inputs and avoids some old Math::BigFloat non-finite results.

    - The legacy nobigint option is now mirrored into XS.
      random_ndigit_prime honors it even when GMP dispatch is available.

    - Better GMP dispatch with big inputs.  is_power, lucas_sequence,
      is_almost_prime, and multifactorial, logint, and rootint avoid older
      GMP paths that could truncate or reject large arguments, while using
      GMP 0.54+ where it supports the larger API.

    - powint handles 0, 1, and -1 with very large exponents correctly.
      Only calls older GMP backends with native-sized exponents.

    - Work around older GMP backends returning empty certificates for 32-bit
      and smaller random Maurer/Shawe-Taylor primes.

    - sieve_range width and depth are validated as native unsigned integers,
      and the depth 0/1 behavior of omitting 0 and 1 is now documented.

    - vecuniq now accepts generic defined inputs using List::Util-like
      string/hash semantics, while retaining a fast native-integer path.

    - allsqrtmod and allrootmod now consistently return the count in scalar
      context.  Documentation and tests added.

    - More list-returning functions consistently return the number of values
      in scalar context.  Documentation and tests added.

    - todigitstring with negative bigint in base 10 included the sign.  Fixed.

    - Powerfree functions now consistently validate k (limited to 2^32-1)
      and document negative-n behavior for count/sum/part variants.

    - Fix nth_powerfree near UV_MAX and keep refinement counts synchronized.

    - rootmod no longer misses some roots modulo prime powers.

    - vecreduce, vecslide, and vecwindow block values are no longer aliases
      of input values in XS, matching the PP implementations and vecpairwise.

    - random_ndigit_prime(3) was undersampling 101 (since v0.62).

    - random_prime and random_ndigit_prime require GMP 0.54+ for direct GMP
      dispatch, preserving uniform selection after the initial random trials.

    - omega_prime_count no longer misses some large 7- and 8-omega primes.
      omega_primes uses bounded memory for large non-prefix ranges.

    - binomialmod avoids native overflow with negative n and very large k,
      and is much faster for small k with large prime-power moduli.

    - Frobenius (P,Q) uses gcd(n,2QD) and checks disc a square, not |disc|.

    - Fix PP Frobenius-Khashin parameter selection, document 2018 algorithm.

    - Real functions avoid intermediate overflow for large finite inputs,
      handle special values correctly, and preserve exact inverse_li results.


    [FUNCTIONALITY AND PERFORMANCE]

    - Some functions now have custom op support.  This means in Perl 5.14+
      they are resolved at Perl's compile time, skipping function overhead.
      Functions like addint, mulint, irand64, etc. are 10-50% faster.

    - Native integer sorting with vecsort/vecsorti detects ordered input and
      skips unnecessary radix passes, especially useful for narrow signed values.

    - vecequal handles circular array references without unbounded recursion.

    - On platforms with support for 128-bit integers, but without the GMP
      backend, we factor 128-bit integers in C.
      This includes 32-bit Perl with modern compilers.
      This can be 100 to 1000 times faster than factoring in Perl.
      In turn this can speed up some other functions that require factoring.
      Installing the GMP backend is preferred.

    - Added native C 64-bit and 128-bit ECM (tinyecm), for much faster
      non-GMP factoring.  PP ECM now uses a deterministic two-stage
      tinyecm-based implementation.

    - PP factoring has improved ECM, p-1, and pbrent implementations.
      General factoring removes the initial pbrent stage, starts with a
      deeper p-1 stage, uses an ECM ladder, and correctly uses HOLF.

    - mertens accepts an optional lower bound.  Native Mertens uses segmented
      Möbius generation and shares calculation state between range endpoints,
      reducing memory and runtime.

    - is_prob_prime, is_bpsw_prime, and is_prime can use native 128-bit
      primality tests on 64-bit builds with uint128_t support.

    - 32-bit XS builds with 128-bit compiler support now use C for sumtotient
      and sum_primes.  Big speedups on large inputs.

    - urandomm is up to 5x faster without GMP.

    - sieve_range in XS uses a partial sieve.  It can be 10x or more faster
      for large native ranges and moderate depths.

    - Tightened prime count bounds (upper/lower) using Dusart 2018 and
      Axler 2022.  Thanks to Pierre Dusart for sending me a copy of his paper.

    - Use amagic calls from XS to perform some integer ops directly using the
      bigint objects rather than calling into Perl.  Streamlined conversion
      of output to canonical form.  1.5 to 3x faster for some examples.

    - simple add/sub/mul/div/pow operations directly on the string input.
      Mostly bypassed if we see Math::GMPz or Math::GMP, but with other classes
      this can be a 2-8x speedup.

    - string bigint accelerations of logint, rootint, and sqrtint
      for 5x to 100x speedup.  absint and negint, can be 5x faster.

    - lshiftint, rshiftint, rashiftint similar, for small shift amounts.

    - vecsum and vecprod 100x faster if GMP not available (vs Calc).

    - znlog faster, especially PP, native prime-power subgroups, and
      large-order rho cases.

    - forsetproduct no longer aliases the input arrays.

    - randperm consistently uses a Fisher-Yates-Knuth shuffle in all cases,
      both in XS and PP.  randperm(n,k) returns the first k values of the
      same process as randperm(n).  A shuffle map is used so the full range
      doesn't always have to be allocated.

    - randperm, shuffle, and vecsample all return the count of results when
      called in scalar context, without consuming random state.

    - Removed many direct PP-to-GMP fallback calls.  Dispatch to the GMP
      backend is done directly from XS in most cases.  This does impact the
      no-XS-but-with-GMP configuration which should only be seen for testing.

    - znorder(a, p^e) is much faster for large prime powers p^e.

    - primes() and twin_primes() can directly dispatch from XS to GMP with
      the GMP 0.54 back end.  This can give a modest speedup.

    - sopf, sopfr, prime_signature, dedekind_psi, aliquot_sum, abundance,
      and is_safe_prime can directly dispatch from XS to GMP with GMP 0.54.

    - binomialmod can directly dispatch to the GMP 0.54 back end, avoiding
      construction of huge intermediate binomial values in more cases.

    - We have a new strint binomial, so large binomials can be computed
      entirely in standard C.  We still prefer the GMP backend.  This is
      much faster than default Math::BigInt.


0.74 2026-03-13

    [API Changes]

    - is_pseudoprime, is_euler_pseudoprime, and is_strong_pseudoprime will
      use an implicit base of 2 rather than a missing base error.  This also
      means is_pseudoprime($n, @bases) works properly.

    - is_strong_pseudoprime passes for bases equal to 0 mod n.  This matches
      what the GMP code always did, and means primes return 1 for any base.

    - divisor_sum(0) returns 0.  divisors(0) returns an empty list.

    - divisors takes an optional second argument to prune the returned list.

    - consecutive_integer_lcm(0) was returning 0.  Now returns 1, matching
      the original GMP API as well as the OEIS series.

    - some error messages with invalid arguments have changed.

    - Callers must 'use Math::Prime::Util::MemFree;' if they want to use it.

    - bernfrac was documented to only accept non-negative inputs.  This is
      now enforced.

    - bernfrac and harmfrac return values are only bigints if needed.

    - The real functions no longer modify their behavior based on whether the
      "bignum" package was loaded, whether in scope or not.

    [ADDED]

    - powersum(n,k)                   sum of k-th powers of positive ints <= n
    - sumpowerful(n[,k])              sum of k-powerful numbers <= n
    - sumliouville(n)                 L(n), sum of liouville(1..n)
    - sumtotient(n)                   sum of euler_phi(1..n)
    - prime_bigomega(n)               # of factors (with multiplicity)
    - prime_omega(n)                  # of factors (distinct)
    - powint(a, b)                    signed integer a^b
    - mulint(a, b)                    signed integer a*b
    - addint(a, b)                    signed integer a+b
    - subint(a, b)                    signed integer a-b
    - add1int(n)                      signed integer n+1
    - sub1int(n)                      signed integer n-1
    - divint(a, b)                    signed integer a/b (floor)
    - modint(a, b)                    signed integer a%b (floor)
    - cdivint(a, b)                   signed integer a/b (ceiling)
    - divrem(a, b)                    Euclidean quotient and remainder
    - fdivrem(a, b)                   Floored quotient and remainder
    - cdivrem(a, b)                   Ceiling quotient and remainder
    - tdivrem(a, b)                   Truncated quotient and remainder
    - lshiftint(n, k)                 left shift n by k bits
    - rshiftint(n, k)                 right shift n by k bits (truncating)
    - rashiftint(n, k)                right shift n by k bits (flooring)
    - absint(n)                       integer absolute value
    - negint(n)                       integer negation
    - cmpint(a, b)                    integer comparison (like <=>)
    - signint(n)                      integer sign (-1,0,1)
    - random_safe_prime               for n-bit safe primes
    - inverse_li_nv(x)                float inverse logarithmic integral
    - is_gaussian_prime(a,b)          is a+bi a Gaussian prime
    - is_lucky(n)                     predicate for lucky number sieve
    - is_happy(n)                     if n happy number, returns height of n
    - is_happy(n,base,exponent)       as above but with given base and power
    - is_smooth(n,k)                  is n a k-smooth number
    - is_rough(n,k)                   is n a k-rough number
    - is_practical(n)                 is n a practical number
    - is_delicate_prime(n[,b])        is n a digitally delicate prime (base b)
    - is_sum_of_squares(n[,k])        can n be a sum of k squares (dflt k=2)
    - is_congruent_number(n)          is n a "congruent number"
    - is_perfect_number(n)            is n equal to sum of its proper divisors
    - is_omega_prime(k,n)             is n divisible by exactly k primes
    - is_almost_prime(k,n)            does n have exactly k prime factors
    - is_chen_prime(n)                is n prime and n+2 prime or semiprime
    - is_powerfree(n[,k])             is n a k-powerfree number (default k=2)
    - is_powerful(n[,k])              is n a k-powerful number (default k=2)
    - is_perfect_power(n)             is n a perfect power (1,4,8,9,16,25,..)
    - is_odd(n)                       is n an odd integer (not divisible by 2)
    - is_even(n)                      is n an even integer (divisible by 2)
    - is_divisible(n,d)               is n exactly divisible by d
    - is_congruent(n,c,d)             is n congruent to c mod d
    - is_qr(a,n)                      is a a quadratic residue mod n
    - is_cyclic(n)                    does n have exactly one group of order n
    - forsquarefreeint {} lo,hi       loop over square-free integers
    - almost_primes(k,[start,]end)    array ref of k-almost-primes
    - almost_prime_count(k,n)         count of integers with exactly k factors
    - almost_prime_count_approx(k,n)  fast approximate k-almost-prime count
    - almost_prime_count_lower(k,n)   fast k-almost-prime count lower bound
    - almost_prime_count_upper(k,n)   fast k-almost-prime count upper bound
    - nth_almost_prime(k,n)           The nth number with exactly k factors
    - nth_almost_prime_approx(k,n)    fast approximate nth k-almost-prime
    - nth_almost_prime_lower(k,n)     fast nth k-almost-prime lower bound
    - nth_almost_prime_upper(k,n)     fast nth k-almost-prime upper bound
    - foralmostprimes {} k,a,b        loop over k-almost-primes
    - omega_primes(k,[start,]end)     array ref of k-omega-primes
    - omega_prime_count(k,n)          count nums divisible by exactly k primes
    - nth_omega_prime(k,n)            The nth number dvsbl by exactly k primes
    - powerful_numbers([lo,]hi[,k])   array ref of k-powerful from lo to hi
    - powerful_count(n[,k])           count of k-powerful numbers <= n
    - nth_powerful(n[,k])             the nth k-powerful number (default k=2)
    - powerfree_count(n[,k])          count of k-powerfree numbers <= n
    - nth_powerfree(n[,k])            The nth k-powerfree number (default k=2)
    - powerfree_sum(n[,k])            sum of k-powerfree numbers <= n
    - powerfree_part(n[,k])           remove excess powers from n
    - powerfree_part_sum(n[,k])       sum of k-powerfree parts for 1 to n
    - squarefree_kernel(n)            integer radical of |n|
    - perfect_power_count([lo,] hi)   count of perfect powers
    - smooth_count(n,k)               count of k-smooth numbers <= n
    - rough_count(n,k)                count of k-rough numbers <= n
    - allsqrtmod(a,n)                 all square roots of a (mod n)
    - qnr(n)                          least quadratic non-residue
    - subfactorial(n)                 count of derangements of n objects
    - fubini(n)                       Fubini (Ordered Bell) number
    - falling_factorial(x,n)          falling factorial
    - rising_factorial(x,n)           rising factorial
    - binomialmod(n,k,m)              fast binomial(n,k) mod m
    - negmod(a, n)                    -a mod n
    - submod(a, b, n)                 a - b mod n
    - muladdmod(a, b, c, n)           a * b + c mod n
    - mulsubmod(a, b, c, n)           a * b - c mod n
    - rootmod(a, k, n)                modular k-th root
    - allrootmod(a,k,n)               all k-th roots of a (mod n)
    - cornacchia(d,n)                 find x,y for x^2 + d * y^2 = n
    - vecequal(\@a, \@b)              compare two array refs for equality
    - tozeckendorf(n)                 Zeckendorf binary string from n
    - fromzeckendorf(str)             n from Zeckendorf binary string
    - lucasu(p,q,k)                   U(p,q)_k
    - lucasv(p,q,k)                   V(p,q)_k
    - lucasuv(p,q,k)                  U(p,q)_k, V(p,q)_k
    - lucasumod(p,q,k,n)              U(p,q)_k mod n
    - lucasvmod(p,q,k,n)              V(p,q)_k mod n
    - lucasuvmod(p,q,k,n)             (U(p,q)_k mod n, V(p,q)_k mod n)
    - pisano_period(n)                period of modular Fibonacci sequence
    - prime_powers([start,] end)      array ref of prime powers
    - next_prime_power(n)             next prime power: p > n
    - prev_prime_power(n)             previous prime power: p < n
    - prime_power_count([start,] end) count of prime powers
    - prime_power_count_approx(n)     fast approximate prime power count
    - prime_power_count_lower(n)      fast prime power count lower bound
    - prime_power_count_upper(n)      fast prime power count upper bound
    - nth_prime_power(n)              the nth prime power
    - nth_prime_power_approx(n)       fast approximate nth prime power
    - nth_prime_power_lower(n)        fast nth prime power lower bound
    - nth_prime_power_upper(n)        fast nth prime power upper bound
    - next_perfect_power(n)           next perfect power: p > n
    - prev_perfect_power(n)           previous perfect power: p < n
    - perfect_power_count([beg,] end) count of perfect powers
    - perfect_power_count_approx(n)   fast approximate perfect power count
    - perfect_power_count_lower(n)    fast perfect power count lower bound
    - perfect_power_count_upper(n)    fast perfect power count upper bound
    - nth_perfect_power(n)            the nth perfect power
    - nth_perfect_power_approx(n)     fast approximate nth perfect power
    - nth_perfect_power_lower(n)      fast nth perfect power lower bound
    - nth_perfect_power_upper(n)      fast nth perfect power upper bound
    - next_chen_prime(n)              next Chen prime: p > n
    - lucky_numbers([start],] end)    array ref of lucky numbers
    - lucky_count([start,] end)       count of lucky numbers
    - lucky_count_approx(n)           fast approximate lucky count
    - lucky_count_lower(n)            fast lucky count lower bound
    - lucky_count_upper(n)            fast lucky count upper bound
    - nth_lucky(n)                    the nth lucky number
    - nth_lucky_approx(n)             fast approximate nth lucky number
    - nth_lucky_lower(n)              fast nth lucky number lower bound
    - nth_lucky_upper(n)              fast nth lucky number upper bound
    - chinese2([a1,m1],[a2,m2],...)   CRT returning (solution,modulus)
    - frobenius_number(...)           Frobenius number of a set
    - vecmex(...)                     least non-negative value not in list
    - vecpmex(...)                    least positive value not in list
    - vecuniq(...)                    remove duplicates from list of integers
    - vecfreq(...)                    return hash of item => count from list
    - vecsingleton(...)               remove all items that aren't unique
    - vecslide { ... } @list          calls block for each pair in list
    - vecsort(@L)                     numerically sort integer list
    - vecsorti(\@L)                   in-place numeric sort a list ref
    - vecsample(k,@list)              return k random elements of list
    - toset(...)                      convert to int set (unique sorted aref)
    - setinsert(\@A,$v)               insert integer v into integer set A
    - setinsert(\@A,\@B)              insert list B into integer set A
    - setremove(\@A,$v)               remove integer v from integer set A
    - setremove(\@A,\@B)              remove list B values from integer set A
    - setinvert(\@A,$v)               if v is in set A, remove, otherwise add
    - setinvert(\@A,\@B)              invert for all values in integer list B
    - setcontains(\@A,...)            are list values all in int set A
    - setcontains(\@A,\@B)            is int set B a subset of int set A
    - setcontainsany(\@A,...)         are any list values in int set A
    - setcontainsany(\@A,\@B)         is any value in B in int set A
    - setbinop {...} \@A[,\@B]        apply op to cross product of set(s)
    - sumset(\@A[,\@B])               new set from a+b {a:A,b:B}
    - setunion(\@A,\@B)               union of two integer lists
    - setintersect(\@A,\@B)           intersection of two integer lists
    - setminus(\@A,\@B)               difference of two integer lists
    - setdelta(\@A,\@B)               symmetric difference of two int lists
    - is_sidon_set(\@L)               is integer list L a Sidon set
    - is_sumfree_set(\@L)             is integer list L a sum-free set
    - set_is_disjoint(\@A,\@B)        is set B disjoint from set A
    - set_is_equal(\@A,\@B)           is set B equal to set A
    - set_is_subset(\@A,\@B)          is set B a subset of set A
    - set_is_proper_subset(\@A,\@B)   is set B a proper subset of set A
    - set_is_superset(\@A,\@B)        is set B a superset of set A
    - set_is_proper_superset(\@A,\@B) is set B a proper superset of set A
    - set_is_proper_intersection(\@A,\@B)   "" a proper intersection ""
    - contfrac(n,d)                   list of continued fraction for n/d
    - from_contfrac(@A)               return (p,q) rational from cfrac list
    - next_calkin_wilf(n,d)           next breadth-first CW rational
    - next_stern_brocot(n,d)          next breadth-first SB rational
    - calkin_wilf_n(n,d)              index of breadth-first CW rational
    - stern_brocot_n(n,d)             index of breadth-first SB rational
    - nth_calkin_wilf(n)              CW rational at breadth-first index n
    - nth_stern_brocot(n)             SB rational at breadth-first index n
    - nth_stern_diatomic(n)           Stern's Diatomic series; fusc(n)
    - farey(n)                        list of Farey sequence order n
    - farey(n,k)                      k'th entry of Farey sequence order n
    - next_farey(n,[p,q])             next order-n rational after p/q
    - farey_rank(n,[p,q])             number of F_n less than p/q
    - minimal_goldbach_pair(n)        least prime p where n-p is also prime
    - goldbach_pair_count(n)          count of how many prime pairs sum to n
    - goldbach_pairs(n)               array of all p where p and n-p are prime


    [FIXES]

    - nth_ramanujan_* returns undef with input of 0.

    - PP code for is_polygonal with n=0 was wrong.  Fixed and refactored.

    - Some large results that used GMP are properly objectified.

    - When using GMP, native int results are no longer objectified.

    - AKS was modding a with r, since 2012.

    - is_totient(2^k) with k >= 33, found by Trizen in 2019.

    - forsquarefree and forfactored were missing a destroy.  From Trizen.

    - todigitstring with non-standard bases and long strings.  From Trizen.

    - Fix v0.62 breaking extended precision long doubles on some machines.
      Big difference in epsilon-level precision for LambertW, Li, Zeta.

    - sieve_range documentation for depth didn't exactly match the XS code.

    - is_extra_strong_lucas_pseudoprime(5) was returning 0.

    - lucas_sequence wasn't right for some outlier cases.  This did not
      impact any primality tests or other internal code.

    - gcd(-n) and lcm(-n) would return -n instead of n.

    - All mod functions are more consistent.  Like Pari, we use mod abs(n).
      If n=0, we return undef.  If n=1, we return 0.

    - invmod(0,1) = 0 instead of undef.  Both cases make sense, but
      Pari, Mathematica, SAGE, and Math::BigInt all return 0.

    - sqrtmod was not solving some cases, e.g. sqrtmod(4,8).

    - chinese uses abs modulos, and values are all pre-modded.
      This fixes negative inputs, e.g. chinese([-4,17],[17,-19])

    - The 70-rt-bignum test was horrendously slow with bignum 0.60+.
      It was bypassing the validation that sanitized the input.

    - Fix logint for base > n.  Github #75.

    - lcm with empty list returns 1 instead of 0.  Github #73.

    - fix an XS stack issue with calling other routines inside forprimes

    - is_pseudoprime and is_euler_pseudoprime have consistent XS, PP, and GMP
      behaviour.  Specifically, single digit inputs and bases a multiple of n.

    - kronecker in XS with 63-bit signed a and 64-bit unsigned n fixed.

    - forfactored and forsquarefree were incorrect near 64-bit boundary.

    - trial_factor etc. take an SV rather than silently coerce to a UV.

    - PP trial_factor corrects for MPU::GMP's different API.

    - PP is_perfect_square was wrong for 18446744073709551521 on 64-bit.

    [FUNCTIONALITY AND PERFORMANCE]

    - LMO prime count is 1.2-1.5x faster.

    - refactor objectification.  Doesn't force.  Enable on Perl < 5.9.

    - save a nanosecond or two in 32-bit primality testing.

    - Rewrite Ramanujan prime bounds using Axler (2017).  For large values
      this is much improved.  Big speedup for large exact nth/count.

    - Approx Ramanujan prime nth and count are much more accurate.

    - Internally, approx prime counts are faster, as we don't need
      to compute RiemannR to extreme precision.

    - Faster and tighter nth_prime_upper bounds.

    - Rewritten C version of Mertens based on pseudocode from Trizen.
      Over 100x faster at 2^32 and grows O(n^(2/3)) vs O(n).

    - fromdigits for bigints uses subquadratic algorithm, thanks from Trizen.
      It also will call GMP if possible.  The large example runs 100x faster.

    - ExponentialIntegral with quadmath is much more precise across the range.

    - rewrote C factorialmod.  1.1-2x faster for small inputs, 3-4x for large.
      The PP non-GMP version also has optimizations added.

    - sum_primes faster for some inputs.

    - Rewrote legendre_phi and removed code duplication.  Much faster.
      Also makes _legendre_pi(n) about 4x faster.

    - Rewrote prime count caching and removed code duplication.
      Uses less memory at large sizes and is faster.

    - Error message "must be a positive integer" changed to
      "must be a non-negative integer" when zero is allowed.

    - More accurate semiprime_count_approx and nth_semiprime_approx.

    - Switched from Kahan to Neumaier sum in some real computations.

    - is_primitive_root faster.  Much faster for p^k and 2p^k with k >= 2.

    - znprimroot faster.  2-20x faster for p^k and 2p^k with k >= 2.

    - native integer input parsing is a little faster.  A big deal for fast
      functions and long lists.

    - rootint, is_power, {next,prev}_perfect_power up to 2x faster.

    - inverse_totient memory leak fixed, memory use reduced, slightly faster.

    - forsetproduct handles tied input arrays.

    - stirling first kind without GMP is faster for some inputs.

    - Block functions that use MULTICALL now will pre-check the block to
      see if it needs to be run in its own scope.  If not, then we get a
      10-30% performance increase with lightweight blocks.

    [OTHER]

    - Legendre, Meissel, and Lehmer prime counting got rewritten to use our
      common code.  3x less code.  No longer buildable standalone.  Included
      by default, though still not called internally.

    - lucas_sequence(n,p,q,k) deprecated.  Use lucasuvmod(p,q,k,n) instead.

    - Documentation more consistent with 'positive' vs 'non-negative'.

    - valuation(n,k) now will error if k < 2.  This follows Pari and SAGE.
      undef is returned for n = 0.  Arguable "Inf" would be preferable.

    - Added ability to select the bigint library, and rewrote large parts of
      the Perl backend to support other bigint classes (e.g. Math::GMPz and
      Math::GMP) rather than being specific to Math::BigInt.

    - Older versions of Test::More with Math::GMPz and Math::GMP have issues
      comparing big integers.  Lots of the tests will stringify the output
      so is(x,y,txt) will work even if x or y is an object.


0.73 2018-11-15

    [ADDED]

    - inverse_totient(n)              the image of euler_phi(n)

    [FIXES]

    - Try to work around 32-bit platforms in semiprime approximations.
      Cannot reproduce on any of my 32-bit test platforms.

    - Fix RT 127605, memory use in for... iterators.


0.72 2018-11-08

    [ADDED]

    - nth_semiprime(n)                the nth semiprime
    - nth_semiprime_approx(n)         fast approximate nth semiprime
    - semiprime_count_approx(n)       fast approximate semiprime count
    - semi_primes                     as primes but for semiprimes
    - forsetproduct {...} \@a,\@b,... Cartesian product of list refs

    [FIXES]

    - Some platforms are extremely slow for is_pillai.  Speed up tests.

    - Ensure random_factored_integer factor list is sorted min->max.

    - forcomposites didn't check lastfor on every callback.

    - Sun's compilers, in a valid interpretation of the code, generated
      divide by zero code for pillai testing.

    [FUNCTIONALITY AND PERFORMANCE]

    - chebyshev_theta and chebyshev_psi redone and uses a table.
      Large inputs are significantly faster.

    - Convert some FP functions to use quadmath if possible.  Without
      quadmath there should be no change.  With quadmath functions like
      LogarithmicIntegral and LambertW will be slower but more accurate.

    - semiprime_count for non-trivial inputs uses a segmented sieve and
      precalculates primes for larger values so can run 2-3x faster.

    - forsemiprimes uses a sieve so large ranges are much faster.

    - ranged moebius more efficient for small intervals.

    - Thanks to GRAY for his module Set::Product which has clean and
      clever XS code, which I used to improve my code.

    - forfactored uses multicall.  Up to 2x faster.

    - forperm, forcomb, forderange uses multicall.  2-3x faster.

    - Frobenius-Khashin algorithm changed from 2013 version to 2016/2018.


0.71 2018-08-28

    [ADDED]

    - forfactored { ... } a,b         loop n=a..b setting $_=n, @_=factor(n)
    - forsquarefree { ... } a,b       as forfactored, but only square-free n
    - forsemiprimes { ... } a,b       as forcomposites, but only semiprimes
    - random_factored_integer(n)      random [1..n] w/ array ref of factors
    - semiprime_count([lo],hi)        counts semiprimes in range

    [FIXES]

    - Monolithic sieves beyond 30*2^32 (~ 1.2 * 10^11) overflowed.

    - is_semiprime was wrong for five small values since 0.69.  Fixed.

    [FUNCTIONALITY AND PERFORMANCE]

    - is_primitive_root much faster (doesn't need to calulate totient,
      and faster rejection when n has no primitive root).

    - znprimroot and znorder use Montgomery, 1.2x to 2x faster.

    - slightly faster sieve_range for native size inputs (use factor_one).

    - bin/primes.pl faster for palindromic primes and works for 10^17

    [OTHER]

    - Added ability to use -DBENCH_SEG for benchmarking sieves using
      prime_count and ntheory::_segment_pi without table optimizations.

    - Reorg of main factor loop.  Should be identical from external view.

    - Internal change to is_semiprime and is_catalan_pseudoprime.


0.70 2017-12-02

    [FIXES]

    - prime_count(a,b) incorrect for a={3..7} and b < 66000000.
      First appeared in v0.65 (May 2017).
      Reported by Trizen.  Fixed.

    - Also impacted were nth_ramanujan_prime and _lower/_upper for
      small input values.

    [FUNCTIONALITY AND PERFORMANCE]

    - Some utility functions used prime counts.  Unlink for more isolation.

    - prime_count_approx uses full precision for bigint or string input.

    - LogarithmicIntegral and ExponentialIntegral will try to use
      our GMP backend if possible.

    - Work around old Math::BigInt::FastCalc (as_int() doesn't work right).

    - prime_memfree also calls GMP's memfree function.  This will clear the
      cached constants (e.g. Pi, Euler).

    - Calling srand or csrand will also result in the GMP backend CSPRNG
      functions being called.  This gives more consistent behavior.

    [OTHER]

    - Turned off threads testing unless release or extended testing is used.
      A few smokers seem to have threads lib that die before we event start.

    - Removed all Math::MPFR code and references.  The latest GMP backend
      has everything we need.

    - The MPU_NO_XS and MPU_NO_GMP environment variables are documented.


0.69 2017-11-08

    [ADDED]

    - is_totient(n)                   true if euler_phi(x) == n for some x

    [FUNCTIONALITY AND PERFORMANCE]

    - is_square_free uses abs(n), like Pari and moebius.

    - is_primitive_root could be wrong with even n on some platforms.

    - euler_phi and moebius with negative range inputs weren't consistent.

    - factorialmod given a large n and m where m was a composite with
      large square factors was incorrect.  Fixed.

    - numtoperm will accept negative k values (k is always mod n!)

    - Split XS mapping of many primality tests.  Makes more sense and
      improves performance for some calls.

    - Split final test in PP cluster sieve.

    - Support some new Math::Prime::Util::GMP functions from 0.47.

    - C spigot Pi is 30-60% faster on x86_64 by using 32-bit types.

    - Reworked some factoring code.

    - Remove ISAAC (Perl and C) since we use ChaCha.

    - Each thread allocs a new const array again instead of sharing.


0.68 2017-10-19

    [API Changes]

    - forcomb with one argument iterates over the power set, so k=0..n
      instead of k=n.  The previous behavior was undocumented.  The new
      behavior matches Pari/GP (forsubset) and Perl6 (combinations).

    [ADDED]

    - factorialmod(n,m)               n! mod m calculated efficiently
    - is_fundamental(d)               true if d a fundamental discriminant

    [FUNCTIONALITY AND PERFORMANCE]

    - Unknown bigint classes no longer return two values after objectify.
      Thanks to Daniel Șuteu for finding this.

    - Using lastfor inside a formultiperm works correctly now.

    - randperm a little faster for k < n cases, and can handle big n
      values without running out of memory as long as k << n.
      E.g. 5000 random native ints without dups:  @r = randperm(~0,5000);

    - forpart with primes pulls min/max values in for a small speedup.

    - forderange 10-20% faster.

    - hammingweight for bigints 3-8x faster.

    - Add Math::GMPq and Math::AnyNum as possible bigint classes.  Inputs
      of these types will be relied on to stringify correctly, and if this
      results in an integer string, to intify correctly.  This should give
      a large speedup for these types.

    - Factoring native integers is 1.2x - 2x faster.  This is due to a
      number of changes.

    - Add Lehman factoring core.  Since this is not exported or used by
      default, the API for factor_lehman may change.

    - All new Montgomery math.  Uses mulredc asm from Ben Buhrow.
      Faster and smaller.  Most primality and factoring code 10% faster.

    - Speedup for factoring by running more Pollard-Rho-Brent, revising
      SQUFOF, updating HOLF, updating recipe.


0.67 2017-09-23

    [ADDED]

    - lastfor                         stops forprimes (etc.) iterations
    - is_square(n)                    returns 1 if n is a perfect square
    - is_polygonal(n,k)               returns 1 if n is a k-gonal number

    [FUNCTIONALITY AND PERFORMANCE]

    - shuffle prototype is @ instead of ;@, so matches List::Util.

    - On Perl 5.8 and earlier we will call PP instead of trying
      direct-to-GMP.  Works around a bug in XS trying to turn the
      result into an object where 5.8.7 and earlier gets lost.

    - We create more const integers, which speeds up common uses of
      permutations.

    - CSPRNG now stores context per-thread rather than using a single
      mutex-protected context.  This speeds up anything using random
      numbers a fair amount, especially with threaded Perls.

    - With the above two optimizations, randperm(144) is 2.5x faster.

    - threading test has threaded srand/irand test added back in, showing
      context is per-thread.  Each thread gets its own sequence and calls
      to srand/csrand and using randomness doesn't impact other threads.


0.66 2017-09-12

    [ADDED]

    - random_semiprime                random n-bit semiprime (even split)
    - random_unrestricted_semiprime   random n-bit semiprime
    - forderange { ... } n            derangements iterator
    - numtoperm(n,k)                  returns kth permutation of n elems
    - permtonum([...])                returns rank of permutation array ref
    - randperm(n[,k])                 random permutation of n elements
    - shuffle(...)                    random permutation of an array

    [FUNCTIONALITY AND PERFORMANCE]

    - Rewrite sieve marking based on Kim Walisch's new simple mod-30 sieve.
      Similar in many ways to my old code, but this is simpler and faster.

    - is_pseudoprime, is_euler_pseudoprime, is_strong_pseudoprime changed to
      better handle the unusual case of base >= n.

    - Speedup for is_carmichael.

    - is_frobenius_underwood_pseudoprime checks for jacobi == 0.  Faster.

    - Updated Montgomery inverse from Robert Gerbicz.

    - Tighter nth prime bounds for large inputs from Axler 2017-06.
      Redo Ramanujan bounds since they're based on nth prime bounds.

    - chinese objectifies result (i.e. big results are bigints).

    - Internal support for Baillie-Wagstaff (pg 1402) extra Lucas tests.

    - More standardized Lucas parameter selection.  Like other tests and the
      1980 paper, checks jacobi(D) in the loop, not gcd(D).

    - entropy_bytes, srand, and csrand moved to XS.

    - Add -secure import to disallow all manual seeding.


0.65 2017-05-03

    [API Changes]

    - Config options irand and primeinc are deprecated.  They will carp if set.

    [FUNCTIONALITY AND PERFORMANCE]

    - Add Math::BigInt::Lite to list of known bigint objects.

    - sum_primes fix for certain ranges with results near 2^64.

    - is_prime, next_prime, prev_prime do a lock-free check for a find-in-cache
      optimization.  This is a big help on on some platforms with many threads.

    - C versions of LogarithmicIntegral and inverse_li rewritten.
      inverse_li honors the documentation promise within FP representation.
      Thanks to Kim Walisch for motivation and discussion.

    - Slightly faster XS nth_prime_approx.

    - PP nth_prime_approx uses inverse_li past 1e12, which should run
      at a reasonable speed now.

    - Adjusted crossover points for segment vs. LMO interval prime_count.

    - Slightly tighter prime_count_lower, nth_prime_upper, and Ramanujan bounds.


0.64 2017-04-17

    [FUNCTIONALITY AND PERFORMANCE]

    - inverse_li switched to Halley instead of binary search.  Faster.

    - Don't call pre-0.46 GMP backend directly for miller_rabin_random.


0.63 2017-04-16

    [FUNCTIONALITY AND PERFORMANCE]

    - Moved miller_rabin_random to separate interface.
      Make catching of negative bases more explicit.


0.62 2017-04-16

    [API Changes]

    - The 'irand' config option is removed, as we now use our own CSPRNG.
      It can be seeded with csrand() or srand().  The latter is not exported.
      This removes the use of Bytes::Random::Secure::Tiny.

    - The 'primeinc' config option is deprecated and will go away soon.

    [ADDED]

    - irand()                  Returns uniform random 32-bit integer
    - irand64()                Returns uniform random 64-bit integer
    - drand([fmax])            Returns uniform random NV (floating point)
    - urandomb(n)              Returns uniform random integer less than 2^n
    - urandomm(n)              Returns uniform random integer in [0, n-1]
    - random_bytes(nbytes)     Return a string of CSPRNG bytes
    - csrand(data)             Seed the CSPRNG
    - srand([UV])              Insecure seed for the CSPRNG (not exported)
    - entropy_bytes(nbytes)    Returns data from our entropy source

    - :rand                    Exports srand, rand, irand, irand64

    - nth_ramanujan_prime_upper(n)       Upper limit of nth Ramanujan prime
    - nth_ramanujan_prime_lower(n)       Lower limit of nth Ramanujan prime
    - nth_ramanujan_prime_approx(n)      Approximate nth Ramanujan prime
    - ramanujan_prime_count_upper(n)     Upper limit of Ramanujan prime count
    - ramanujan_prime_count_lower(n)     Lower limit of Ramanujan prime count
    - ramanujan_prime_count_approx(n)    Approximate Ramanujan prime count

    [FUNCTIONALITY AND PERFORMANCE]

    - vecsum is faster when returning a bigint from native inputs (we
      construct the 128-bit string in C, then call _to_bigint).

    - Add a simple Legendre prime sum using uint128_t, which means only for
      modern 64-bit compilers.  It allows reasonably fast prime sums for
      larger inputs, e.g. 10^12 in 10 seconds.  Kim Walisch's primesum is
      much more sophisticated and over 100x faster.

    - is_pillai about 10x faster for composites.

    - Much faster Ramanujan prime count and nth prime.  These also now use
      vastly less memory even with large inputs.

    - small speed ups for cluster sieve.

    - faster PP is_semiprime.

    - Add prime option to forpart restrictions for all prime / non-prime.

    - is_primitive_root needs two args, as documented.

    - We do random seeding ourselves now, so remove dependency.

    - Random primes functions moved to XS / GMP, 3-10x faster.


0.61 2017-03-12

    [ADDED]

    - is_semiprime(n)        Returns 1 if n has exactly 2 prime factors
    - is_pillai(p)           Returns 0 or v wherev v! % n == n-1 and n % v != 1
    - inverse_li(n)          Integer inverse of Logarithmic Integral

    [FUNCTIONALITY AND PERFORMANCE]

    - is_power(-1,k) now returns true for odd k.

    - RiemannZeta with GMP was not subtracting 1 from results > 9.

    - PP Bernoulli algorithm changed to Seidel from Brent-Harvey.  2x speedup.
      Math::BigNum is 10x faster, and our GMP code is 2000x faster.

    - LambertW changes in C and PP.  Much better initial approximation, and
      switch iteration from Halley to Fritsch.  2 to 10x faster.

    - Try to use GMP LambertW for bignums if it is available.

    - Use Montgomery math in more places:
       = sqrtmod.  1.2-1.7x faster.
       = is_primitive_root.  Up to 2x faster for some inputs.
       = p-1 factoring stage 1.

    - Tune AKS r/s selection above 32-bit.

    - primes.pl uses twin_primes function for ~3x speedup.

    - native chinese can handle some cases that used to overflow.  Use Shell
      sort on moduli to prevent pathological-but-reasonable test case.

    - chinese directly to GMP

    - Switch to Bytes::Random::Secure::Tiny -- fewer dependencies.

    - PP nth_prime_approx has better MSE and uses inverse_li above 10^12.

    - All random prime functions will use GMP versions if possible and
      if a custom irand has not been configured.
      They are much faster than the PP versions at smaller bit sizes.

    - is_carmichael and is_pillai small speedups.


0.60 2016-10-09

    [ADDED]

    - vecfirstidx { expr } @n             returns first index with expr true

    [FUNCTIONALITY AND PERFORMANCE]

    - Expanded and modified prime count sparse tables. Prime counts from 30k
      to 90M are 1.2x to 2.5x faster.  It has no appreciable effect on the
      speed of prime counts larger than this size.

    - fromdigits works with bigint first arg, no need to stringify.
      Slightly faster for bigints, but slower than desired.

    - Various speedups and changes for fromdigits, todigits, todigitstring.

    - vecprod in PP for negative high-bit would return double not bigint.

    - Lah numbers added as Stirling numbers of the third kind.  They've been
      in the GMP code for almost 2 years now.  Also for big results, directly
      call the GMP code and objectify the result.

    - Small performance change to AKS (r,s) selection tuning.

    - On x86_64, use Montgomery math for Pollard/Brent Rho.  This speeds up
      factoring significantly for large native inputs (e.g. 10-20 digits).

    - Use new GMP zeta and riemannr functions if possible, making some of
      our operations much faster without Math::MPFR.

    - print_primes with large args will try GMP sieve for big speedup.  E.g.
        use bigint;  print_primes(2e19,2e19+1e7);
      goes from 37 minutes to 7 seconds.  This also removes a mistaken blank
      line at the end for certain ranges.

    - PP primes tries to use GMP.  Only for calls from other PP code.

    - Slightly more accuracy in native ExponentialIntegral.

    - Slightly more accuracy in twin_prime_count_approx.

    - nth_twin_prime_approx was incorrect over 1e10 and over 2e16 would
      infinite loop due to Perl double conversion.

    - nth_twin_prime_approx a little faster and more accurate.


0.59 2016-08-03

    [ADDED]

    - is_euler_plumb_pseudoprime  Plumb's Euler Criterion test.
    - is_prime_power              Returns k if n=p^k for p a prime.
    - logint(n,b)                 Integer logarithm.  Largest e s.t. b^e <= n.
    - rootint(n,k)                Integer k-th root.
    - ramanujan_sum(k,n)          Ramanujan's sum

    [FUNCTIONALITY AND PERFORMANCE]

    - Fixes for quadmath:
      + Fix "infinity" in t/11-primes.t.
      + Fix native Pi to use quads.
      + Trim some threading tests.

    - Fix fromdigits memory error with large string.

    - Remove 3 threading tests that were causing issues with Perl -DDEBUGGING.

    - foroddcomposites with some odd start values could index incorrectly.

    - is_primitive_root(1,0) returns 0 instead of fp exception.

    - mertens() uses a little less memory.

    - 2x speedup for znlog with bigint values.

    - is_pseudoprime() and is_euler_pseudoprime() use Montgomery math so are
      much faster.  They seem to be ~5% faster than Miller-Rabin now.

    - is_catalan_pseudoprime 1.1x to 1.4x faster.

    - is_perrin_pseudoprime over 10x faster.
      Uses Adams/Shanks doubling and Montgomery math.
      Single core, odd composites: ~8M range/s.

    - Add restricted Perrin pseudoprimes using an optional argument.

    - Add bloom filters to reject non-perfect cubes, fifths, and sevenths.
      is_power about 2-3x faster for native inputs.

    - forcomposites / foroddcomposites about 1.2x faster past 64-bit.

    - exp_mangoldt rewritten to use is_prime_power.

    - Integer root code rewritten and now exported.

    - We've been hacking around the problem of older Perls autovivifying
      functions at compile time.  This makes functions that don't exist
      return true when asked if they're defined, which causes us distress.

      Store the available GMP functions before loading the PP code.

      XS code knows MPU::GMP version and calls as appropriate.  This works
      around the auto-vivication, and lets us choose to call the GMP
      function based on version instead of just existence.
      E.g. GMP's is_power was added in 0.19, but didn't support negative
      powers until 0.28.


0.58 2016-05-21

    [API Changes]

    - prev_prime($n) where $n <= 2 now returns undef instead of 0.  This
      may enable catching range errors, and is technically more correct.

    - nth_prime(0) now returns undef instead of 0.  This should help catch
      cases where the base wasn't understood.  The change is similar for
      all the nth_* functions (e.g. nth_twin_prime).

    - sumdigits(n,base) will interpret n as a number in the given base,
      rather than the Pari/GP method of converting decimal n to that base
      then summing.  This allows sumdigits to easily sum hex strings.
      The old behavior is easily done with vecsum(todigits(n, base)).

    - binary() was not intended to be released (todigits and todigitstring
      are supersets), but the documentation got left in.  Remove docs.

    [ADDED]

    - addmod(a, b, n)                     a + b mod n
    - mulmod(a, b, n)                     a * b mod n
    - divmod(a, b, n)                     a / b mod n
    - powmod(a, b, n)                     a ^ b mod n
    - sqrtmod(a, n)                       modular square root
    - is_euler_pseudoprime(n,a[...])      Euler test to given bases
    - is_primitive_root(r, n)             is r a primitive root mod n
    - is_quasi_carmichael(n)              is n a Quasi-Carmichael number
    - hclassno(n)                         Hurwitz class number H(n) * 12
    - sieve_range(n, width, depth)        sieve to given depth, return offsets

    [FUNCTIONALITY AND PERFORMANCE]

    - Fixed incorrect table entries for 2^16th Ramanujan prime count and
      nth_ramanujan_prime(23744).

    - foroddcomposites with certain arguments would start with 10 instead of 9.

    - lucasu and lucasv should return bigint types.

    - vecsum will handle 128-bit sums internally (performance increase).

    - Speedup is_carmichael.

    - Speedup znprimroot, 10% for small inputs, 10x for large composites.

    - Speedup znlog ~2x.  It is now Rho racing an interleaved BSGS.

    - Change AKS to Bernstein 2003 theorem 4.1.
      5-20x faster than Bornemann, 20000+x faster than V6.

    - sum_primes now uses tables for native sizes (performance increase).

    - ramanujan_tau uses Cohen's hclassno method instead of the sigma
      calculation.  This is 3-4x faster than the GMP code for inputs > 300k,
      and much faster than the older PP code.

    - fromdigits much faster for large base-10 arrays.  Timing is better than
      split plus join when output is a bigint.


0.57 2016-01-03

    [ADDED]

    - formultiperm { ... } \@n            loop over multiset permutations
    - todigits(n[,base[,len]])            convert n to digit array
    - todigitstring(n[,base[,len]])       convert n to string
    - fromdigits(\@d[,base])              convert digit array ref to number
    - fromdigits(str[,base])              convert string to number
    - ramanujan_prime_count               counts Ramanujan primes in range
    - vecany { expr } @n                  true if any expr is true
    - vecall { expr } @n                  true if all expr are true
    - vecnone { expr } @n                 true if no expr are true
    - vecnotall { expr } @n               true if not all expr are true
    - vecfirst { expr } @n                returns first element with expr true

    [FUNCTIONALITY AND PERFORMANCE]

    - nth_ramanujan_prime(997) was wrong.  Fixed.

    - Tighten Ramanujan prime bounds.  Big speedups for large nth Rp.


0.56 2015-12-13

    [ADDED]

    - is_carmichael(n)                    Returns 1 if n is a Carmichael number
    - forcomp { ... } n[,{...}]           loop over compositions

    [FUNCTIONALITY AND PERFORMANCE]

    - Faster, nonrecursive divisors_from_factors routine.

    - gcdext(0,0) returns (0,0,0) to match GMP and Pari/GP.

    - Use better prime count lower/upper bounds from Büthe 2015.

    - forpart and forcomp both use lexicographic order (was anti-lexico).


0.55 2015-10-19

    - Fixed test that was using a 64-bit number on 32-bit machines.

    [FUNCTIONALITY AND PERFORMANCE]

    - Speed up PP versions of sieve_prime_cluster, twin_primes,
      twin_prime_count, nth_twin_prime, primes.


0.54 2015-10-14

    [ADDED]

    - sieve_prime_cluster(low,high[,...]) find prime clusters

    [Misc]

    - Certain small primes used to return false with Frobenius and AES Lucas
      tests when given extra arguments.  Both are unusual cases never used
      by the main system.  Fixed.


0.53 2015-09-05

    [ADDED]

    - ramanujan_tau(n)                    Ramanujan's Tau function
    - sumdigits(n[,base])                 sum digits of n

    [FUNCTIONALITY AND PERFORMANCE]

    - Don't use Math::MPFR unless underlying MPFR library is at least 3.x.

    - Use new Math::Prime::Util::GMP::sigma function for divisor_sum.

    - Use new Math::Prime::Util::GMP::sieve_twin_primes(a,b).


0.52 2015-08-09

    [ADDED]

    - is_square_free(n)                   Check for repeated factors

    [FUNCTIONALITY AND PERFORMANCE]

    - print_primes with 2 args was sending to wrong fileno.

    - Double speed of sum_primes.

    - Rewrote some internal sieve-walking code, speeds up next_prime,
      forprimes, print_primes, and more.

    - Small speedup for forcomposites / foroddcomposites.

    - Small speedup for is_prime with composite 32+ bit inputs.

    - is_frobenius_khashin_pseudoprime now uses Montgomery math for speed.

    - PrimeArray now treats skipping forward by relatively small amounts as
      forward iteration.  This makes it much more efficient for many cases,
      but does open up some pathological cases.

    - PrimeArray now allows exporting @primes (and a few others), which
      saves some typing.

    - PrimeArray now works for indices up to 2^32-1, after which it silently
      rolls over.  Previously it worked to 2^31-1 then croaked.

    - PrimeIterator now uses small segments instead of always next_prime.
      A little more memory, but 2-4x faster.

    - factor, divisor, fordivisors and some others should better keep
      bigint types (e.g. Math::GMPz input yields Math::GMPz output).

    - Faster GCD on some platforms.

    - Peter Dettman supplied a patch for Shawe-Taylor prime generation to
      make it deterministically match reference implementations.  Thanks!

    [Misc]

    - Check for old MPFR now using C library version, not module version.

    - prime_count_{lower,upper} now uses MPFR to give full precision.

    - Montgomery math and uint128_t enabled on Darwin/clang.


0.51 2015-06-21

    [ADDED]

    - sum_primes(lo,hi)                   Summation of primes in range
    - print_primes(lo,hi[,fd])            Print primes to stdout or fd
    - is_catalan_pseudoprime(n)           Catalan primality test
    - is_frobenius_khashin_pseudoprime(n) Khashin's 2013 Frobenius test

    [FUNCTIONALITY AND PERFORMANCE]

    - Slightly faster PP sieving using better code from Perlmonks.

    - Lucas sequence works with even valued n.

    - Used idea from Colin Wright to speed up is_perrin_pseudoprime 5x.
      We can check smaller congruent sequences for composites as a prefilter.

    - is_frobenius_pseudoprime no longer checks for perfect squares, and
      doesn't bail to BPSW if P,Q,D exceed n.  This makes it produce some
      pseudoprimes it did not before (but ought to have).

    [Misc]

    - Work with old MPFR (some test failures in older Win32 systems).

    - Don't assert in global destructor if a MemFree object is destroyed.


0.50 2015-05-03

    [ADDED]

    - harmfrac(n)               (num,den) of Harmonic number
    - harmreal(n)               Harmonic number as BigFloat
    - sqrtint(n)                Integer square root of n
    - vecextract(\@arr, mask)   Return elements from arr selected by mask
    - ramanujan_primes(lo,hi)   Ramanujan primes R_n in [lo,hi]
    - nth_ramanujan_prime(n)    the nth Ramanujan prime R_n
    - is_ramanujan_prime(n)     1 if n is a Ramanujan prime, 0 otherwise

    [FUNCTIONALITY AND PERFORMANCE]

    - Implement single-base hashed M-R for 32-bit inputs, inspired by
      Forišek and Jančina 2015 as well as last year's tests with
      2-base (2^49) and 3-base (2^64) hashed solutions for MPU.  Primality
      testing is 20-40% faster for this size.

    - Small speedups for znlog.

    - PP nth_prime on 32-bit fixed for values over 2^32.

    [Misc]

    - Changes to nth_prime_{lower,upper}.  They use the Axler (2013) bounds,
      and the XS code will also use inverse prime count bounds for small
      values.  This gives 2-10x tighter bounds.

    - Tighten prime count bounds using Axler, Kotnik, Büthe.  Thanks to
      Charles R Greathouse IV for pointing me to these.


0.49  2014-11-30

    - Make versions the same in all packages.


0.48  2014-11-28

    [ADDED]

    - lucasu(P, Q, k)           U_k for Lucas(P,Q)
    - lucasv(P, Q, k)           V_k for Lucas(P,Q)

    [Misc]

    - Use Axler (2014) bounds for prime count where they improve on Dusart.


0.47  2014-11-18

    [ADDED]

    - is_mersenne_prime(p)      returns 1 iff 2^p-1 is prime

    [FUNCTIONALITY AND PERFORMANCE]

    - Standalone compilation (e.g. factoring without Perl installed) is easier.

    - For next_prime and prev_prime with bigints, stay in XS as long as
      possible to cut overhead.  Up to 1.5x faster.

    - Factoring on 64-bit platforms is faster for 32-bit inputs.

    - AKS is faster for larger than half-word inputs, especially on 64-bit
      machines with gcc's 128-bit types.

    - is_provable_prime goes through XS first, so can run *much* faster for
      small inputs.

    [OTHER]

    - NetBSD improperly exports symbols in string.h, including popcount.
      Rename our internal function to work around it.

    - is_power now takes an optional scalar reference third argument which
      will be set to the root if found.  It also works for negative n.

    - Changes to trim a little memory use.  lucas_sequence goes from
      PP->[XS,GMP,PP] to XS[->PP[->GMP]].  ecm_factor is moved out of root.
      Moved some primality proving logic out of root.

    - primes.pl when given one argument will show primes up to that number.


0.46  2014-10-21

    [API Changes]

    - is_pseudoprime has the same signature as is_strong_pseudoprime now.
      This means it requires one or more bases and has no default base.
      The documentation had never mentioned the default, so this should
      have little impact, and the common signature makes more sense.

    [ADDED]

    - hammingweight(n)          Population count (count binary 1s)
    - vecreduce {...} @v        Reduce/fold, exactly like List::Util::reduce

    [Misc]

    - Syntax fix from Salvatore.

    - vecmin / vecmax in XS, if overflows UV do via strings to avoid PP.

    - Add example for verifying prime gaps, similar to Nicely's cglp4.

    - divisor_sum wasn't running XS code for k=0.  Refactor PP code,
      includes speedup when input is a non-Math::BigInt (e.g. Math::GMP).

    - Improve test coverage.

    [PP Updates]

    - Large speedup for divisors with bigints in 64-100 bit range.

    - Revamp RiemannZeta.  Fixes some bignum output, but requires RT fixes.

    - Optimization for PP comparison to ~0.

    - PP factoring is faster, especially for small inputs.


0.45  2014-09-26

    [ADDED]

    - forcomb { ... } n, k      combinations iterator
    - forperm { ... } n         permutations iterator
    - factorial(n)              n!
    - is_bpsw_prime(n)          primality test with no pretests, just ES BPSW
    - is_frobenius_pseudoprime  Frobenius quadratic primality test
    - is_perrin_pseudoprime     Perrin primality test (unrestricted)
    - vecmin(@list)             minimum of list of integers
    - vecmax(@list)             maximum of list of integers
    - vecprod(@list)            product of list of integers
    - bernfrac(n)               (num,den) of Bernoulli number
    - bernreal(n)               Bernoulli number as BigFloat
    - stirling(n,m,[type])      Stirling numbers of first or second kind
    - LambertW(k)               Solves for W in k = W*exp(W)
    - Pi([digits])              Pi as NV or with requested digits

    [FUNCTIONALITY AND PERFORMANCE]

    - znorder algorithm changed from Das to Cohen for ~1% speedup.

    - factoring sped up a bit for 15-19 digits.

    - speedup for divisor_sum with very large exponents.

    [OTHER]

    - Alias added for the module name "ntheory".  The module has grown
      enough that it seems more appropriate.

    - Big build change: Try a GMP compilation and add Math::Prime::Util::GMP
      to dependency list if it succeeds.

    - Fixed a memory leak in segment_primes / segment_twin_primes introduced
      in previous release.  Thanks Valgrind!


0.43  2014-08-16

    [ADDED]

    - foroddcomposites      like forcomposites, but skips even numbers
    - twin_primes           as primes but for twin primes
    - config: use_primeinc  allow the fast but bad PRIMEINC random prime method

    [REMOVED DEPRECATED NAMES]

    - all_factors         replaced in 0.36 by divisors
    - miller_rabin        replaced in 0.10 by is_strong_pseudoprime

    [FUNCTIONALITY AND PERFORMANCE]

    - Divisors sorted with qsort instead of Shell sort.  No appreciable time
      difference, but slightly less code size.

    - Added Micali-Schnorr generator to examples/csrand.pl.  Made a version
      of csrand that uses Math::GMP for faster operation.

    - Added synopsis release test.  Thanks to Neil Bowers and Toby Inkster.

    - ranged euler_phi is more efficient when lo < 100.

    - factor for 49 to 64-bit numbers sped up slightly (a small p-1 is tried
      before SQUFOF for these sizes).

    - HOLF factoring sped up using premultiplier first.


0.42  2014-06-18

    [ADDED]

    - gcdext(x,y)                            extended Euclidian algorithm
    - chinese([a,n],[a,n],...)               Chinese Remainder

    [FUNCTIONALITY AND PERFORMANCE]

    - znlog is *much* faster.  Added BSGS for XS and PP, Rho works better.

    - Another inverse improvement from W. Izykowski, doing 8 bits at a time.
      A further 1% to 15% speedup in primality testing.

    - A 35% reduction in overhead for forprimes with multicall.

    - prime segment sieving over large ranges will use larger segment sizes
      when given large bases.  This uses some more memory, but is much faster.

    - An alternate method for calculating RiemannR used when appropriate.

    - RiemannZeta caps at 10M even with MPFR.  This has over 300k leading 0s.

    - RiemannR will use the C code if not a BigFloat or without bignum loaded.
      The C code should only take a few microseconds for any value.

    - Refactor some PP code: {next,prev}_prime, chebyshev_{theta,psi}.
      In addition, PP sieving uses less memory.

    - Accelerate nth_twin_prime using the sparse twin_prime_count table.


0.41  2014-05-18

    [ADDED]

    - valuation(n,k)                         how many times does k divide n?
    - invmod(a,n)                            inverse of a modulo n
    - forpart { ... } n[,{...}]              loop over partitions (Pari 2.6.x)
    - vecsum(...)                            sum list of integers
    - binomial(n,k)                          binomial coefficient

    [FUNCTIONALITY AND PERFORMANCE]

    - Big speedup for primality testing in range ~2^25 to 2^64, which also
      affects functions like next_prime, prev_prime, etc.  This is due to two
      changes in the Montgomery math section -- an improvement to mont_prod64
      and using a new modular inverse from W. Izykowski based on Arazi (1994).

    - factoring small inputs (n < 20M) is ~10% faster, which speeds up some
      misc functions (e.g. euler_phi, divisor_sum) for small inputs.

    - Small improvement to twin_prime_count_approx and nth_twin_prime_approx.

    - Better AKS testing in xt/primality-aks.pl.

    - Loosen requirements of lucas_sequence.  More useful for general seqs.
      Add tests for some common sequences.

    - forcomposites handles beg and end near ~0.


0.40  2014-04-21

    [ADDED]

    - random_shawe_taylor_prime              FIPS 186-4 random proven prime
    - random_shawe_taylor_prime_with_cert    as above with certificate
    - twin_prime_count                       counts twin primes in range
    - twin_prime_count_approx                fast approximation to Pi_2(n)
    - nth_twin_prime                         returns the nth twin prime
    - nth_twin_prime_approx                  estimates the nth twin prime

    [FUNCTIONALITY AND PERFORMANCE]

    - Update PP Frobenius-Underwood test.

    - Speed up exp_mangoldt.

    - nth_prime_approx uses inverse RiemannR in XS code for better accuracy.
      Cippola 1902 is still used for PP and large values, with a slightly
      more accurate third order correction.

    - Tighten nth_prime_lower and nth_prime_upper for very small values.

    - Fix legendre_phi when given tiny x and large a (odd test case).
      Some speedups for huge a, from R. Andrew Ohana.

    - Ranged totient is slightly faster with start of 0.

    - Fix random_prime with a bigint prime high value.


0.39  2014-03-01

    - Changed logl to log in AKS.  Critical for FreeBSD and NetBSD.

    - Make sure we don't use Math::BigInt::Pari in threading tests.
      threads + Math::Pari = segfault on UNIX and Windows.

    - Various minor changes trying to guess what ActiveState is doing.


0.38  2014-02-28

    [ADDED]

    - is_power         Returns max k if n=p^k.  See Pari 2.4.x.

    [FUNCTIONALITY AND PERFORMANCE]

    - Factoring powers (and k*n^m for small k) is much faster.

    - Speed up znprimroot.

    - Add Bernstein+Voloch improvements to AKS.  Much faster than the v6
      implementation, though still terribly slow vs. BPSW or other proofs.

    [OTHER]

    - Added some Project Euler examples.

    - If using a threaded Perl without EXTENDED_TESTING, thread tests will
      print diagnostics instead of failing.  This might help find issues
      with platforms that are currently failing with no indications, and
      allow installation for non-threaded use.


0.37  2014-01-26

    [FUNCTIONALITY AND PERFORMANCE]

    - Simplified primes().  No longer takes an optional hashref as first arg,
      which was awkward and never documented.

    - Dynamically loads the PP code and Math::BigInt only when needed.  This
      removes a lot of bloat for the usual cases:

        2.0 MB  perl -E 'say 1'
        4.2 MB  MPU 0.37
        4.5 MB  Math::Prime::XS + Math::Factor::XS
        5.3 MB  Math::Pari
        7.6 MB  MPU 0.34
        9.6 MB  MPU 0.36
        9.7 MB  MPU 0.35

    - Combined with the above, this reduces startup overhead a lot (~3x).

    - Adjusted factor script to lower startup costs.  Over 2x faster
      with native integer (non-expression) arguments.  This is just not
      loading thousands of lines of Perl code that aren't used, which
      was more time-consuming than the actual factoring.

    - nth_prime_{lower,upper,approx} and prime_count_{lower,upper,approx}
      moved to XS->PP.  This helps us slim down and cut startup overhead.

    - Fix doc for znlog:   znlog(a,g,p) finds k s.t. a = g^k mod p


0.36  2014-01-13

    [API Changes]

    - factor behavior for 0 and 1 more consistent.  The results for
      factor, factor_exp, divisors, and divisor_sum now match Pari,
      and the omega(1)/Omega(1) exception is removed.

      Thanks to Hugo van der Sanden for bringing this up.

    - all_factors changed to divisors.  The old name still remains aliased.

    [ADDED]

    - forcomposites    like forprimes, but on composites.  See Pari 2.6.x.
    - fordivisors      calls a block for each divisor
    - kronecker        Kronecker symbol (extension of Jacobi symbol)
    - znprimroot       Primitive root modulo n
    - gcd              Greatest common divisor
    - lcm              Least common multiple
    - legendre_phi     Legendre's sum

    [FUNCTIONALITY AND PERFORMANCE]

    - Win32 fixes from bulk88 / bulkdd.  Thanks!

    - XS redundancy removal and fixes from bulk88 and leont.  Smaller DLL.
      This almost includes not compiling a number of prime count methods
      (Legendre, Meissel, Lehmer, and LMOS) that are not used.  Using
      "-DLEHMER" in the Makefile will compile them, but there should not
      be any reason to do so.

    - Big XS interface reorg.  Most functions now go straight to XS, which
      reduces their overhead.  Input number validation is much faster for
      the general case.  Those two combined meant the '-nobigint' import
      no longer serves any good purpose.

    - More functions will go from XS directly to the GMP module, bypassing
      the Perl layer entirely.  The upside is less overhead, both for the
      case of having GMP, and without.  In the contrived case of having XS
      turned off but the GMP module enabled, things will run slower since
      they no longer go to GMP.

    - Test suite should run faster.  Combination of small speedups to hot
      spots as well as pushing a few slow tasks to EXTENDED_TESTING (these
      are generally things never used, like pure Perl AKS).

    - Some 5.6.2-is-broken workarounds.

    - Some LMO edge cases:  small numbers that only show up if a #define is
      changed, and counts > 18440000000000000000.  Tested through 2^64-1 now.

    - LMO much faster if -march=native is used for gcc on a platform with
      asm popcnt (e.g. Nahalem+, Barcelona+, ARM Neon, SPARC, Power7, etc.).

    - divisors (all_factors) faster for native size numbers with many factors.

    - Switch from mapes to a cached primorial/totient small phi method in
      lehmer.c.  Significant for LMOS and Legendre (no longer used or compiled,
      see earlier.  Thanks to Kim Walisch for questioning my earlier decision.

    - Rewrite sieve composite map.  Segment sieving is faster.  It's a little
      faster than primegen for me, but still slower than primesieve and yafu.

    - znorder uses Carmichael Lambda instead of Euler Phi.  Faster.

    - While Math::BigInt has the bgcd and blcm functions, they are slow for
      native numbers, even with the Pari or GMP back ends.  The gcd/lcm here
      are 20-100x faster.  LCM also returns results consistent with Pari.

    - Removed the old SQUFOF code, so the racing version is the only one.  It
      was already the only one being used.


0.35  2013-12-08

    [API Changes]

    - We now use Math::BigInt in the module rather than dynamically loading
      it, and will switch to BigInts as needed.  The most noticeable effect
      of this is that next_prime() / prev_prime() will switch between BigInt
      and native int at the boundary without regard to the input type or
      whether bigint is in effect, and next_prime will never return 0.
      Additionally, all functions will convert large decimal number strings
      to BigInts if needed.

      $pref = primes("1000000000000000000000", "1000000000000000000999");
      is_prime("882249208105452588824618008529");
      $a = euler_phi("801294088771394680000412");

    [FUNCTIONALITY AND PERFORMANCE]

    - Switched to extended LMO algorithm for prime_count.  Much better memory
      use and much faster for large values.  Speeds up nth_prime also.  Huge
      thanks to Christian Bau's excellent paper and examples.

    - Some fixes for 32-bit.

    - prime_count_approx, upper, and lower return exact answers in more cases.

    - Fixed a problem with Lehmer prime_count introduced in 0.34.

    - nth_prime changed from RiemannR to inverse Li (with partial addition).
      This makes some of the big nth_prime calculations (e.g. 10^15, 10^16)
      run quite a bit faster as they sieve less on average.


0.34  2013-11-19

    - Fixed test that was using a 64-bit number on 32-bit machines.

    - Switch a couple internal arrays from UV to uint32 in prime count.
      This reduces memory consumption a little with big counts.  Total
      memory use for counts > 10^15 is about 5x less than in version 0.31.


0.33  2013-11-18

    [API Changes]

    - all_factors now includes 1 and n, making it identical to Pari's
      divisors(n) function, but no longer identical to Math::Factor::XS's
      factors(n) function.  This change allows consistency between
      divisor_sum(n,0) and scalar all_factors(n).

    [ADDED]

    - factor_exp    returns factors as ([p,e],[p,e],...)
    - liouville     -1^(Omega(n)), OEIS A008836
    - partitions    partition function p(n), OEIS A000041

    [FUNCTIONALITY AND PERFORMANCE]

    - all_factors in scalar context returns sigma_0(n).

    - exp_mangoldt defaults to XS for speed.

    - Fixed Pure Perl 33- to 64-bit is_pseudoprime.

    - prime_count uses Lehmer below a threshold (8000M), LMO above.
      This keeps good performance while still using low memory.  A small
      speedup for small (3-6M) inputs has been added.  Overall memory use
      has been reduced by 2-4x for large inputs.

    - Perl RiemannZeta changes:

      - Borwein Zeta calculations done in BigInt instead of BigFloat (speed).

      - Patch submitted for the frustrating Math::BigFloat defect RT 43692.
        With the patch applied, we get much, much better accuracy.

      - Accuracy updates, especially with fixed BigFloat.

    - Lucas sequence called with bigints will return bigint objects.

    - prime_iterator_object should now work with Iterator::Simple.

    - chebyshev_theta and chebyshev_psi use segmented sieves.

    - More aggressive pruning of tests with 64-bit Perl 5.6.  I'd like to
      just kill support for systems that can't even add two numbers
      correctly, but too many other modules want 5.6 support, and lots of
      our functionality *does* work (e.g. primes, prime count, etc.).


0.32  2013-10-13

    [ADDED]

    - is_provable_prime
    - is_provable_prime_with_cert
    - carmichael_lambda
    - znorder
    - prime_iterator_object
    - miller_rabin_random

    [NEW FEATURES]

    - Added Math::Prime::Util::PrimeIterator.  A more feature-rich iterator
      than the simple closure one from prime_iterator.  Experimental.

    - Make very simple LMO primecount work, and switch prime_count to use it.
      It is slower for large inputs, but uses much less memory.  For smaller
      inputs it it as fast or faster.  Lehmer code modified to constrain
      memory use at the expense of speed (starts taking effect at ~ 10^16).
      Thanks to Kim Walisch for discussions about this.  Note that this is
      a very simple implementation -- better code could run 10x faster and
      use even less memory.

    - divisor_sum can take an integer 'k' in the second argument to compute
      sigma_k.  This is much faster than using subs, especially when the
      result can be computed in XS using native precision.  For integer
      second arguments, the result will automatically be a bigint if needed.
      It is also much faster for larger inputs.

    - factor() can be called in scalar context to give the number of
      prime factors.  The XS function was ignoring the context, and now
      is more consistent.  It also slightly speeds up looking at the
      number of factors, e.g. Omega(x) A001222.

    [FUNCTIONALITY AND PERFORMANCE]

    - Use MPU::GMP::pn_primorial if we have it.

    - Input validation accepts bigint objects and converts them to scalars
      entirely in XS.

    - random_nbit_prime now uses Fouque and Tibouchi A1 for 65+ bits.
      Slightly better uniformity and typically a bit faster.

    - Incorporate Montgomery reduction for primality testing, thanks to
      Wojciech Izykowski.  This is a 1.3 to 1.5x speedup for is_prob_prime,
      is_prime, and is_strong_pseudoprime for numbers > 2^32 on x86_64.
      This also help things like prime_iterator for values > 2^32.

    - Montgomery reduction used in Lucas and Frobenius tests.  Up to 2x
      speedup for 33 to 64-bit inputs on x86_64/gcc platforms.

    - Some fixes around near maxint primes, forprimes, etc.  Includes
      more workarounds for Math::BigInt::GMP's constructor sign bug.

    - Bytes::Random::Secure is loaded only when random prime functionality
      is used.  Shaves a few milliseconds and bytes off of startup.

    - Speedups for Perl (no GMP) primality and random primes.

    [MISC]

    - Primality functions moved to their own file primality.c.

0.31  2013-08-07

    - Change proof certificate documentation to reflect the new text format.

    - Some platforms were using __int128 when it wasn't supported.  Only
      x86_64 and Power64 use it now.

    - Small speedup for ranged totient internals.

    - Patch MPU::GMP 0.13 giving us not quite what we expected from a small
      certificate.  Fixed in MPU::GMP 0.14, worked around here regardless.

0.30  2013-08-06

    [API Changes]
      - Primality proofs now use the new "MPU Certificate" format, which is
        text rather than a nested Perl data structure.  This is much better
        for external interaction, especially with non-Perl tools.  It is
        not quite as convenient for all-Perl manipulation.

    [Functions Added]
      - is_frobenius_underwood_pseudoprime
      - is_almost_extra_strong_lucas_pseudoprime
      - lucas_sequence
      - pplus1_factor

    [Enhancements]
      - Documentation and PP is_prime changed to use extra strong Lucas test
        from the strong test.  This matches what the newest MPU::GMP does.
        This has no effect at all for numbers < 2^64.  No counter-example is
        known for the standard, strong, extra strong, or almost extra strong
        (increment 1 or 2) tests.  The extra strong test is faster than the
        strong test and produces fewer pseudoprimes.  It retains the residue
        class properties of the strong Lucas test (where the SPSP-2
        pseudoprimes favor residue class 1 and the Lucas pseudoprimes favor
        residue class -1), hence should retain the BPSW test strength.

      - XS code for all 4 Lucas tests.

      - Clean up is_prob_prime, also ~10% faster for n >= 885594169.

      - Small mulmod speedup for non-gcc/x86_64 platforms, and for any platform
        with gcc 4.4 or newer.

    [Bug Fixes]
      - Fixed a rare refcount / bignum / callback issue in next_prime.

0.29  2013-05-30

    [Functions Added]
       - is_pseudoprime (Fermat probable prime test)
       - is_lucas_pseudoprime (standard Lucas-Selfridge test)
       - is_extra_strong_lucas_pseudoprime (Mo/Jones/Grantham E.S. Lucas test)

    - Fix a signed vs. unsigned char issue in ranged moebius.  Thanks to the
      Debian testers for finding this.

    - XS is_prob_prime / is_prime now use a BPSW-style test (SPRP2 plus
      extra strong Lucas test) for values over 2^32.  This results in up
      to 2.5x faster performance for large 64-bit values on most machines.
      All PSP2s have been verified with Jan Feitsma's database.

    - forprimes now uses a segmented sieve.  This (1) allows arbitrary 64-bit
      ranges with good memory use, and (2) allows nesting on threaded perls.

    - prime_count_approx for very large values (> 10^36) was very slow without
      Math::MPFR.  Switch to Li+correction for large values if Math::MPFR is
      not available.

    - Workaround for MSVC compiler.

0.28  2013-05-23

    - An optimization to nth_prime caused occasional threaded Win32 faults.
      Adjust so this is avoided.

    - Yet another XS micro-speedup (PERL_NO_GET_CONTEXT)

    - forprimes { block } [begin,]end.  e.g.
        forprimes { say } 100;
        $sum = 0;  forprimes { $sum += $_ } 1000,50000;  say $sum;
        forprimes { say if is_prime($_+2) } 10000;  # print twin primes

    - my $it = prime_iterator(10000); say $it->();
      This is experimental (that is, the interface may change).

0.27  2013-05-20

    - is_prime, is_prob_prime, next_prime, and prev_prime now all go straight
      to XS if possible.  This makes them much faster for small inputs without
      having to use the -nobigint flag.

    - XS simple number validation to lower function call overhead.  Still a
      lot more overhead compared to directly calling the XS functions, but
      it shaves a little bit of time off every call.

    - Speedup pure Perl factoring of small numbers.

    - is_prob_prime / is_prime about 10% faster for composites.

    - Allow '+N' as the second parameter to primes.pl.  This allows:
          primes.pl 100 +30
      to return the primes between 100 and 130.  Or:
          primes.pl 'nth_prime(1000000000)' +2**8

    - Use EXTENDED_TESTING to turn on extra tests.

0.26  2013-04-21

    [Pure Perl Factoring]
        - real p-1 -- much faster and more effective
        - Fermat (no better than HOLF)
        - speedup for pbrent
        - simple ECM
        - redo factoring mix

    [Functions Added]
        prime_certificate  produces a certificate of primality.
        verify_prime       checks a primality certificate.

    - Pure perl primality proof now uses BLS75 instead of Lucas, so some
      numbers will be much faster [n-1 only needs factoring to (n/2)^1/3].

    - Math::Prime::Util::ECAffinePoint and ECProjectivePoint modules for
      dealing with elliptic curves.

0.25  2013-03-19

    - Speed up p-1 stage 2 factoring.  Combined with some minor changes to the
      general factoring combination, ~20% faster for 19 digit semiprimes.

    - New internal macro to loop over primary sieve starting at 2.  Simplifies
      code in quite a few places.

    - Forgot to skip one of the tests with broken 5.6.2.

0.24  2013-03-10

    - Fix compilation with old pre-C99 strict compilers (decl after statement).

    - euler_phi on a range wasn't working right with some ranges.

    - More XS prime count improvements to speed and space.  Add some tables
      to the sieve count so it runs a bit faster.  Transition from sieve later.

    - PP prime count for 10^9 and larger is ~2x faster and uses much less
      memory.  Similar impact for nth_prime 10^8 or larger.

    - Let factor.pl accept expressions just like primes.pl.

0.23  2013-03-05

    - Replace XS Zeta for x > 5 with series from Cephes.  It is 1 eps more
      accurate for a small fraction of inputs.  More importantly, it is much
      faster in range 5 < x < 10.  This only affects non-integer inputs.

    - PP Zeta code replaced (for no-MPFR, non-bignums) with new series.  The
      new code is much more accurate for small values, and *much* faster.

    - Add consecutive_integer_lcm function, just like MPU::GMP's (though we
      define ci_lcm(0) = 0, which should get propogated).

    - Implement binary search on RiemannR for XS nth_prime when n > 2e11.
      Runs ~2x faster for 1e12, 3x faster for 1e13.  Thanks to Programming
      Praxis for the idea and motivation.

    - Add the first and second Chebyshev functions (theta and psi).

    - put isqrt(n) in util.h, use it everywhere.
      put icbrt(n) in lehmer.h, use it there.

    - Start on Lagarias-Miller-Odlyzko prime count.

    - A new data structure for the phi(x,a) function used by all the fast
      prime count routines.  Quite a bit faster and most importantly, uses
      half the memory of the old structure.

    [Performance]
       - Divisor sum with no sub is ~10x faster.
       - Speed up PP version of exp_mangoldt, create XS version.
       - Zeta much faster as mentioned above.
       - faster nth_prime as mentioned above.
       - AKS about 10% faster.
       - Unroll a little more in sieve inner loop.  A couple percent faster.
       - Faster prime_count and nth_prime due to new phi(x,a) (about 1.25x).

0.22  2013-02-26

    - Move main factor loop out of xs and into factor.c.

    - Totient and Moebius now have complete XS implementations.

    - Ranged totient uses less memory when segmented.

    - Switch thread locking to pthreads condition variables.

0.21  2013-02-22

    - Switch to using Bytes::Random::Secure for random primes.  This is a
      big change in that it is the first non-CORE module used.  However, it
      gets rid of lots of possible stupidness from system rand.

    - Spelling fixes in documentation.

    - primes.pl: Add circular and Panaitopol primes.

    - euler_phi and moebius now will compute over a range.

    - Add mertens function: 1000+ times faster than summing moebius($_).

    - Add exp_mangoldt function: exponential of von Mangoldt's function.

    - divisor_sum defaults to sigma if no sub is given (i.e. it sums).

    [Performance]
       - Speedup factoring small numbers.  With -nobigint factoring from
         1 to 10M, it's 1.2x faster.  1.5x faster than Math::Factor::XS.
       - Totient and Möbius over a range are much faster than separate calls.
       - divisor_sum is 2x faster.
       - primes.pl is much faster with Pillai primes.
       - Reduce overhead in euler_phi -- about 2x faster for individual calls.

0.20  2013-02-03

    - Speedup for PP AKS, and turn off test on 32-bit machines.

    - Replaced fast sqrt detection in PP.pm with a slightly slower version.
      The bloom filter doesn't work right in 32-bit Perl.  Having a non-working
      detector led to really bad performance.  Hence this and the AKS change
      should speed up testing on some 32-bit machines by a huge amount.

    - Fix is_perfect_power in XS AKS.

0.19  2013-02-01

    - Update MR bases with newest from http://miller-rabin.appspot.com/.

    - Fixed some issues when using bignum and Calc BigInt backend, and bignum
      and Perl 5.6.

    - Added tests for bigint is_provable_prime.

    - Added a few tests to give better coverage.

    - Adjust some validation subroutines to cut down on overhead.

0.18  2013-01-14

    - Add random_strong_prime.

    - Fix builds with Solaris 9 and older.

    - Add some debug info to perhaps find out why old ActiveState Perls are
      dying in Math::BigInt::Calc, as if they were using really old versions
      that run out of memory trying to calculate '2 ** 66'.
      http://code.activestate.com/ppm/Math-Prime-Util/

0.17  2012-12-20

    - Perl 5.8.1 - 5.8.7 miscalculates 12345 ** 4, which I used in a test.

    - Fix (hopefully) for MSC compilation.

    - Unroll sieve loop for another 20% or so speedup.  It won't have much
      practical application now that we use Lehmer's method for counts, but
      there are some cases that can still show speedups.

    - Changed the rand functionality yet again.  Sorry.  This should give
      better support for plugging in crypto RNG's when used from other
      modules.

0.16  2012-12-11

    - randbits >= 32 on some 32-bit systems was messing us up.  Restrict our
      internal randbits to wordsize-1.

0.15  2012-12-09

    [Enhancements to Ei, li, Zeta, R functions]
       - Native Zeta and R have slightly more accurate results.
       - For bignums, use Math::MPFR if possible.  MUCH faster.
         Also allows extended precision while still being fast.
       - Better accuracy for standard bignums.
       - All four functions do:
          - XS if native input.
          - MPFR to whatever accuracy is desired, if Math::MPFR installed.
          - BigFloat versions if no MPFR and BigFloat input.
          - standard version if no MPFR and not a BigFloat.

    [Other Changes]
    - Add tests for primorial, jordan_totient, and divisor_sum.

    - Revamp of the random_prime internals.  Also fixes some issues with
      random n-bit and maurer primes.

    - The random prime and primorial functions now will return a Math::BigInt
      object if the result is greater than the native size.  This includes
      loading up the Math::BigInt library if necessary.

0.14  2012-11-29

    [Compilation / Test Issues]
       - Fix compilation on NetBSD
       - Try to fix compilation on Win32 + MSVC
       - Speed up some testing, helps a lot with Cygwin on slow machines
       - Speed up a lot of slow PP areas, especially used by test suite

    [Functions Added]
       - jordan_totient          generalization of Euler Totient
       - divisor_sum             run coderef for every divisor

    [Other Changes]
    - XS AKS extended from half-word to full-word.

    - Allow environment variables MPU_NO_XS and MPU_NO_GMP to turn off XS and
      GMP support respectively if they are defined and equal to 1.

    - Lehmer prime count for Pure Perl code, including use in nth_prime.
         prime count 10^9 using sieve:
            71.9s   PP sieve
             0.47s  XS sieve
         prime count 10^9 using Lehmer:
             0.70s  PP lehmer
             0.03s  XS lehmer

    - Moved bignum Zeta and R to separate file, only loaded when needed.
      Helpful to get the big rarely-used tables out of the main loading.

    - Quote arguments to Math::Big{Int,Float} in a few places it wasn't.
      Math::Big* coerces the input to a signed value if it isn't a string,
      which causes us all sorts of grief.

0.13  2012-11-19

    - Fix an issue with prime count, and make prime count available as a
      standalone program using primesieve.

0.12  2012-11-17

    [Programs Added]
       - bin/primes.pl
       - bin/factor.pl

    [Functions Added]
       - primorial               product of primes <= n
       - pn_primorial            product of first n primes
       - prime_set_config        set config options
       - RiemannZeta             export and make accurate for small reals
       - is_provable_prime       prove primes after BPSW
       - is_aks_prime            prove prime via AKS

    [Other Changes]
    - Add 'assume_rh' configuration option (default: false) which can be set
      to allow functions to assume the Riemann Hypothesis.

    - Use the Schoenfeld bound for Pi(x) (x large) if assume_rh is true.

    - valgrind testing

    - Use long doubles for math functions.

    - Some fixes and speedups for ranged primes().

    - In the PP code, use 2 MR bases for more numbers when possible.

    - Fixup of racing SQUFOF, and switch to use it in factor().

    - Complete rewrite of XS p-1 factor routine, includes second stage.

    - bug fix for prime_count on edge of cache.

    - prime_count will use Lehmer prime counting algorithm for largish
      sizes (above 4 million).  This is MUCH faster than sieving.

    - nth_prime now uses the fast Lehmer prime count below the lower limit,
      then sieves up from there.  This makes a big speed difference for inputs
      over 10^6 or so -- over 100x faster for 10^9 and up.

0.11  2012-07-23
    - Turn off threading tests on Cygwin, as threads on some Cygwin platforms
      give random panics (my Win7 64-bit works fine, XP 32-bit does not).
    - Use pow instead of exp2 -- some systems don't have exp2.
    - Fix compile issues on MSC, thanks to Sisyphus.
    - some bigint/bignum changes (next_prime and math functions).
    - speed up and enhance some tests.
    - Test version of racing SQUFOF (not used in main code yet).  Also add
      a little more up-front trial division for main factor routine.


0.10  2012-07-16
    - full bigint support for everything.  Use '-nobigint' as an import to
      shortcut straight to XS for better speed on some of the very fast functions.
      This involved moving a lot of functions into Util.pm.

    - added BPSW primality test for large (>2^64) is_prob_prime and is_prime.

    - Add tests for pp and bignum, cleanup of many tests.

    - New bounds for prime_count and nth_prime.  Dusart 2010 for larger
      values, tuned nth_prime_upper for small values.  Much tighter.

    [Functions Added]
       - prime_get_config              to get configuration options
       - is_strong_pseudoprime         better name for miller_rabin
       - is_strong_lucas_pseudoprime   strong lucas-selfridge psp test
       - random_nbit_prime             for n-bit primes
       - random_maurer_prime           provable n-bit primes
       - moebius                       Mo:bius function
       - euler_phi                     Euler's phi aka totient

    [Minor Changes]
       - Make miller_rabin return 0 if given even number.
       - The XS miller_rabin code now works with large base > n.
       - factor always returns sorted results
       - miller_rabin() deprecated.  Use is_strong_pseudoprime instead.

    [Support all functionality of:]
       - Math::Prime::XS         (MPU: more functions, a bit faster)
       - Math::Prime::FastSieve  (MPU: more functions, a bit faster)
       - Math::Prime::TiedArray  (MPU: a *lot* faster)
       - Math::Factor::XS        (MPU: bignums, faster, missing multiplicity)
       - Math::Big::Factors      (MPU: orders of magnitude faster)
       - Math::Primality         (MPU: more portable, fast native, slow bigint)
                                 (MPU+MPU::GMP: faster)
       - Crypt::Primes           (MPU: more portable, slower & no fancy options)

    [Support some functionality of:]
       - Math::Big               (MPU's primes is *much* faster)
       - Bit::Vector             (MPU's primes is ~10x faster)

0.09  2012-06-25
    - Pure Perl code added.  Passes all tests.  Used only if the XSLoader
      fails.  It's 1-120x slower than the C code.  When forced to use the
      PP code, the test suite is 38x slower on 64-bit, 16x slower on 32-bit
      (in 64-bit, the test suite runs some large numbers through routines
      like prime_count and nth_prime that are much faster in C).
    - Modifications to threading test:
        - some machines were failing because they use non-TS rand.  Fix by
          making our own rand.
        - Win32 was failing because of unique threading issues.  It barfs
          if you free memory on a different thread than allocated it.
    - is_prime could return 1 in some cases.  Fixed to only return 0 or 2.

0.08  2012-06-22
    - Added thread safety and tested good concurrency.
    - Accuracy improvement and measurements for math functions.
    - Remove simple sieve -- it wasn't being used, and was just around for
      performance comparisons.
    - Static presieve for 7, 11, and 13.  1k of ROM used for prefilling sieve
      memory, meaning we can skip the 7, 11, and 13 loops.  ~15% speedup.
    - Add all_factors function and added tests to t/50-factoring.t.
    - Add tied array module Math::Prime::Util::PrimeArray.
    - 5.6.2 64-bit now disables the 64-bit factoring tests instead of failing
      the module.  The main issue is that we can't verify the factors since Perl
      can't properly multiply them.

0.07  2012-06-17
    - Fixed a bug in next_prime found by Lou Godio (thank you VERY much!).
      Added more tests for this.  This had been changed in another area but
      hadn't been brought into next_prime.

0.06  2012-06-14
    - Change to New/Safefree from malloc.  Oops.

0.05  2012-06-11
    - Speed up mulmod: asm for GCC + x86_64, native 64-bit for 32-bit Perl
      is uint64_t is available, and range tests for others.  This speeds up
      some of the factoring as well as Miller-Rabin, which in turn speeds up
      is_prime.  is_prime is used quite commonly, so this is good.
    - nth_prime routines should now all croak on overflow in the same way.
    - Segmented prime_count, things like this are reasonably efficient:
            say prime_count( 10**16,  10**16 + 2**20 )
    - Add Ei(x), li(x), and R(x) functions.
    - prime_count_approx uses R(x), making it vastly more accurate.
    - Let user override rand for random_prime.
    - Add many more tests with the help of Devel::Cover.

0.04  2012-06-07
    - Didn't do tests on 32-bit machine before release.  Test suite caught
      problem with next_prime overflow.
    - Try to use 64-bit modulo math even when Perl is 32-bit.  It can make
      is_prime run up to 10x faster (which impacts next_prime, factoring, etc.)
    - replace all assert with croak indicating an internal error.
    - Add random_prime and random_ndigit_prime
    - renamed prime_free to prime_memfree.

0.03  2012-06-06
    - Speed up factoring.
    - fixed powmod routine, speedup for smaller numbers
    - Add Miller-Rabin and deterministic probable prime functions.  These
      are now used for is_prime and factoring, giving a big speedup for
      numbers > 32-bit.
    - Add HOLF factoring (just for demo)
    - Next prime returns 0 on overflow

0.02  2012-06-05
    - Back off new_ok to new/isa_ok to keep Test::More requirements low.
    - Some documentation updates.
    - I accidently used long in SQUFOF, which breaks LLP64.
    - Test for broken 64-bit Perl.
    - Fix overflow issues in segmented sieving.
    - Switch to using UVuf for croaks.  What I should have done all along.
    - prime_count uses a segment sieve with 256k chunks (~7.9M numbers).
      Not memory intensive any more, and faster for large inputs.  The time
      growth is slightly over linear however, so expect to wait a long
      time for 10^12 or more.
    - nth_prime also transitioned to segmented sieve.

0.01  2012-06-04
    - Initial release
