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

0.37  2014-01-26


    - 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.


    - 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


    - 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");
      $a = euler_phi("801294088771394680000412");


    - 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).


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


    - 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


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


    - 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.


    - 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.


    - 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

      - 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  This allows:
 100 +30
      to return the primes between 100 and 130.  Or:
 '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 accept expressions just like

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.

       - 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.

    - 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).

       - 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.
       - 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 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

    - 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'.

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

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/
       - bin/

    [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

    - 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