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Revision history for Perl module Math::Prime::Util
0.46 20141021
[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 nonMath::BigInt (e.g. Math::GMP).
 Improve test coverage.
[PP Updates]
 Large speedup for divisors with bigints in 64100 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 20140926
[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 1519 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 20140816
[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 MicaliSchnorr 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 64bit numbers sped up slightly (a small p1 is tried
before SQUFOF for these sizes).
 HOLF factoring sped up using premultiplier first.
0.42 20140618
[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 20140518
[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/primalityaks.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 20140421
[ADDED]
 random_shawe_taylor_prime FIPS 1864 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 FrobeniusUnderwood 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 20140301
 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 20140228
[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 nonthreaded use.
0.37 20140126
[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 (nonexpression) arguments. This is just not
loading thousands of lines of Perl code that aren't used, which
was more timeconsuming 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 20140113
[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 5.6.2isbroken workarounds.
 Some LMO edge cases: small numbers that only show up if a #define is
changed, and counts > 18440000000000000000. Tested through 2^641 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 20100x 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 20131208
[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 32bit.
 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 20131119
 Fixed test that was using a 64bit number on 32bit 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 20131118
[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 64bit 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 (36M) inputs has been added. Overall memory use
has been reduced by 24x 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 64bit 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 20131013
[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 featurerich 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 64bit 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 20130807
 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 20130806
[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 nonPerl tools. It is
not quite as convenient for allPerl 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 counterexample 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 SPSP2
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 nongcc/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 20130530
[Functions Added]
 is_pseudoprime (Fermat probable prime test)
 is_lucas_pseudoprime (standard LucasSelfridge 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 BPSWstyle test (SPRP2 plus
extra strong Lucas test) for values over 2^32. This results in up
to 2.5x faster performance for large 64bit values on most machines.
All PSP2s have been verified with Jan Feitsma's database.
 forprimes now uses a segmented sieve. This (1) allows arbitrary 64bit
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 20130523
 An optimization to nth_prime caused occasional threaded Win32 faults.
Adjust so this is avoided.
 Yet another XS microspeedup (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 20130520
 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 20130421
[Pure Perl Factoring]
 real p1  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 [n1 only needs factoring to (n/2)^1/3].
 Math::Prime::Util::ECAffinePoint and ECProjectivePoint modules for
dealing with elliptic curves.
0.25 20130319
 Speed up p1 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 20130310
 Fix compilation with old preC99 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 20130305
 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 noninteger inputs.
 PP Zeta code replaced (for noMPFR, nonbignums) 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 LagariasMillerOdlyzko 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 20130226
 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 20130222
 Switch to using Bytes::Random::Secure for random primes. This is a
big change in that it is the first nonCORE 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 MÃ¶bius 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 20130203
 Speedup for PP AKS, and turn off test on 32bit machines.
 Replaced fast sqrt detection in PP.pm with a slightly slower version.
The bloom filter doesn't work right in 32bit Perl. Having a nonworking
detector led to really bad performance. Hence this and the AKS change
should speed up testing on some 32bit machines by a huge amount.
 Fix is_perfect_power in XS AKS.
0.19 20130201
 Update MR bases with newest from http://millerrabin.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 20130114
 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/MathPrimeUtil/
0.17 20121220
 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 20121211
 randbits >= 32 on some 32bit systems was messing us up. Restrict our
internal randbits to wordsize1.
0.15 20121209
[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 nbit 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 20121129
[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 halfword to fullword.
 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 rarelyused 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 20121119
 Fix an issue with prime count, and make prime count available as a
standalone program using primesieve.
0.12 20121117
[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 p1 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 20120723
 Turn off threading tests on Cygwin, as threads on some Cygwin platforms
give random panics (my Win7 64bit works fine, XP 32bit 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 upfront trial division for main factor routine.
0.10 20120716
 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 lucasselfridge psp test
 random_nbit_prime for nbit primes
 random_maurer_prime provable nbit 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 20120625
 Pure Perl code added. Passes all tests. Used only if the XSLoader
fails. It's 1120x slower than the C code. When forced to use the
PP code, the test suite is 38x slower on 64bit, 16x slower on 32bit
(in 64bit, 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 nonTS 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 20120622
 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/50factoring.t.
 Add tied array module Math::Prime::Util::PrimeArray.
 5.6.2 64bit now disables the 64bit 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 20120617
 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 20120614
 Change to New/Safefree from malloc. Oops.
0.05 20120611
 Speed up mulmod: asm for GCC + x86_64, native 64bit for 32bit Perl
is uint64_t is available, and range tests for others. This speeds up
some of the factoring as well as MillerRabin, 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 20120607
 Didn't do tests on 32bit machine before release. Test suite caught
problem with next_prime overflow.
 Try to use 64bit modulo math even when Perl is 32bit. 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 20120606
 Speed up factoring.
 fixed powmod routine, speedup for smaller numbers
 Add MillerRabin and deterministic probable prime functions. These
are now used for is_prime and factoring, giving a big speedup for
numbers > 32bit.
 Add HOLF factoring (just for demo)
 Next prime returns 0 on overflow
0.02 20120605
 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 64bit 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 20120604
 Initial release
