```
Revision history for Perl module Math::Prime::Util
0.59 2016-08-03
[ADDED]
- 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
```