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| ntheory(3) | User Contributed Perl Documentation | ntheory(3) |
NAME
ntheory - Number theory utilitiesSEE
See Math::Prime::Util for complete documentation.QUICK REFERENCE
Tags: :all to import almost all functions :rand to import rand, srand, irand, irand64PRIMALITY
is_prob_prime(n) primality test (BPSW) is_prime(n) primality test (BPSW + extra) is_provable_prime(n) primality test with proof is_provable_prime_with_cert(n) primality test: (isprime,cert) prime_certificate(n) as above with just certificate verify_prime(cert) verify a primality certificate is_mersenne_prime(p) is 2^p-1 prime or composite is_aks_prime(n) AKS deterministic test (slow) is_ramanujan_prime(n) is n a Ramanujan prime
PROBABLE PRIME TESTS
is_pseudoprime(n,bases) Fermat probable prime test is_euler_pseudoprime(n,bases) Euler test to bases is_euler_plumb_pseudoprime(n) Euler Criterion test is_strong_pseudoprime(n,bases) Miller-Rabin test to bases is_lucas_pseudoprime(n) Lucas test is_strong_lucas_pseudoprime(n) strong Lucas test is_almost_extra_strong_lucas_pseudoprime(n, [incr]) AES Lucas test is_extra_strong_lucas_pseudoprime(n) extra strong Lucas test is_frobenius_pseudoprime(n, [a,b]) Frobenius quadratic test is_frobenius_underwood_pseudoprime(n) combined PSP and Lucas is_frobenius_khashin_pseudoprime(n) Khashin's 2013 Frobenius test is_perrin_pseudoprime(n [,r]) Perrin test is_catalan_pseudoprime(n) Catalan test is_bpsw_prime(n) combined SPSP-2 and ES Lucas miller_rabin_random(n, ntests) perform random-base MR tests
PRIMES
primes([start,] end) array ref of primes twin_primes([start,] end) array ref of twin primes semi_primes([start,] end) array ref of semiprimes ramanujan_primes([start,] end) array ref of Ramanujan primes sieve_prime_cluster(start, end, @C) list of prime k-tuples sieve_range(n, width, depth) sieve out small factors to depth next_prime(n) next prime > n prev_prime(n) previous prime < n prime_count(n) count of primes <= n prime_count(start, end) count of primes in range prime_count_lower(n) fast lower bound for prime count prime_count_upper(n) fast upper bound for prime count prime_count_approx(n) fast approximate count of primes nth_prime(n) the nth prime (n=1 returns 2) nth_prime_lower(n) fast lower bound for nth prime nth_prime_upper(n) fast upper bound for nth prime nth_prime_approx(n) fast approximate nth prime twin_prime_count(n) count of twin primes <= n twin_prime_count(start, end) count of twin primes in range twin_prime_count_approx(n) fast approx count of twin primes nth_twin_prime(n) the nth twin prime (n=1 returns 3) nth_twin_prime_approx(n) fast approximate nth twin prime semiprime_count(n) count of semiprimes <= n semiprime_count(start, end) count of semiprimes in range semiprime_count_approx(n) fast approximate count of semiprimes nth_semiprime(n) the nth semiprime nth_semiprime_approx(n) fast approximate nth semiprime ramanujan_prime_count(n) count of Ramanujan primes <= n ramanujan_prime_count(start, end) count of Ramanujan primes in range ramanujan_prime_count_lower(n) fast lower bound for Ramanujan count ramanujan_prime_count_upper(n) fast upper bound for Ramanujan count ramanujan_prime_count_approx(n) fast approximate Ramanujan count nth_ramanujan_prime(n) the nth Ramanujan prime (Rn) nth_ramanujan_prime_lower(n) fast lower bound for Rn nth_ramanujan_prime_upper(n) fast upper bound for Rn nth_ramanujan_prime_approx(n) fast approximate Rn legendre_phi(n,a) # below n not div by first a primes inverse_li(n) integer inverse logarithmic integral prime_precalc(n) precalculate primes to n sum_primes([start,] end) return summation of primes in range print_primes(start,end[,fd]) print primes to stdout or fd
FACTORING
factor(n) array of prime factors of n
factor_exp(n) array of [p,k] factors p^k
divisors(n) array of divisors of n
divisor_sum(n) sum of divisors
divisor_sum(n,k) sum of k-th power of divisors
divisor_sum(n,sub{...}) sum of code run for each divisor
znlog(a, g, p) solve k in a = g^k mod p
ITERATORS
forprimes { ... } [start,] end loop over primes in range
forcomposites { ... } [start,] end loop over composites in range
foroddcomposites {...} [start,] end loop over odd composites in range
forsemiprimes {...} [start,] end loop over semiprimes in range
forfactored {...} [start,] end loop with factors
forsquarefree {...} [start,] end loop with factors of square-free n
fordivisors { ... } n loop over the divisors of n
forpart { ... } n [,{...}] loop over integer partitions
forcomp { ... } n [,{...}] loop over integer compositions
forcomb { ... } n, k loop over combinations
forperm { ... } n loop over permutations
formultiperm { ... } \@n loop over multiset permutations
forderange { ... } n loop over derangements
forsetproduct { ... } \@a[,...] loop over Cartesian product of lists
prime_iterator returns a simple prime iterator
prime_iterator_object returns a prime iterator object
lastfor stop iteration of for.... loop
RANDOM NUMBERS
irand random 32-bit integer irand64 random 64-bit integer drand([limit]) random NV in [0,1) or [0,limit) random_bytes(n) string with n random bytes entropy_bytes(n) string with n entropy-source bytes urandomb(n) random integer less than 2^n urandomm(n) random integer less than n csrand(data) seed the CSPRNG with binary data srand([seed]) simple seed (exported with :rand) rand([limit]) alias for drand (exported with :rand) random_factored_integer(n) random [1..n] and array ref of factors
RANDOM PRIMES
random_prime([start,] end) random prime in a range random_ndigit_prime(n) random prime with n digits random_nbit_prime(n) random prime with n bits random_strong_prime(n) random strong prime with n bits random_proven_prime(n) random n-bit prime with proof random_proven_prime_with_cert(n) as above and include certificate random_maurer_prime(n) random n-bit prime w/ Maurer's alg. random_maurer_prime_with_cert(n) as above and include certificate random_shawe_taylor_prime(n) random n-bit prime with S-T alg. random_shawe_taylor_prime_with_cert(n) as above including certificate random_unrestricted_semiprime(n) random n-bit semiprime random_semiprime(n) as above with equal size factors
LISTS
vecsum(@list) integer sum of list
vecprod(@list) integer product of list
vecmin(@list) minimum of list of integers
vecmax(@list) maximum of list of integers
vecextract(\@list, mask) select from list based on mask
vecreduce { ... } @list reduce / left fold applied to list
vecall { ... } @list return true if all are true
vecany { ... } @list return true if any are true
vecnone { ... } @list return true if none are true
vecnotall { ... } @list return true if not all are true
vecfirst { ... } @list return first value that evals true
vecfirstidx { ... } @list return first index that evals true
MATH
todigits(n[,base[,len]]) convert n to digit array in base todigitstring(n[,base[,len]]) convert n to string in base fromdigits(\@d,[,base]) convert base digit vector to number fromdigits(str,[,base]) convert base digit string to number sumdigits(n) sum of digits, with optional base is_square(n) return 1 if n is a perfect square is_power(n) return k if n = c^k for integer c is_power(n,k) return 1 if n = c^k for integer c, k is_power(n,k,\$root) as above but also set $root to c. is_prime_power(n) return k if n = p^k for prime p is_prime_power(n,\$p) as above but also set $p to p is_square_free(n) return true if no repeated factors is_carmichael(n) is n a Carmichael number is_quasi_carmichael(n) is n a quasi-Carmichael number is_primitive_root(r,n) is r a primitive root mod n is_pillai(n) v where v! % n == n-1 and n % v != 1 is_semiprime(n) does n have exactly 2 prime factors is_polygonal(n,k) is n a k-polygonal number is_polygonal(n,k,\$root) as above but also set $root is_fundamental(d) is d a fundamental discriminant is_totient(n) is n = euler_phi(x) for some x sqrtint(n) integer square root rootint(n,k) integer k-th root rootint(n,k,\$rk) as above but also set $rk to r^k logint(n,b) integer logarithm logint(n,b,\$be) as above but also set $be to b^e. gcd(@list) greatest common divisor lcm(@list) least common multiple gcdext(x,y) return (u,v,d) where u*x+v*y=d chinese([a,mod1],[b,mod2],...) Chinese Remainder Theorem primorial(n) product of primes below n pn_primorial(n) product of first n primes factorial(n) product of first n integers: n! factorialmod(n,m) factorial mod m binomial(n,k) binomial coefficient partitions(n) number of integer partitions valuation(n,k) number of times n is divisible by k hammingweight(n) population count (# of binary 1s) kronecker(a,b) Kronecker (Jacobi) symbol 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 invmod(a,n) inverse of a modulo n sqrtmod(a,n) modular square root moebius(n) Moebius function of n moebius(beg, end) array of Moebius in range mertens(n) sum of Moebius for 1 to n euler_phi(n) Euler totient of n euler_phi(beg, end) Euler totient for a range inverse_totient(n) image of Euler totient jordan_totient(n,k) Jordan's totient carmichael_lambda(n) Carmichael's Lambda function ramanujan_sum(k,n) Ramanujan's sum exp_mangoldt exponential of Mangoldt function liouville(n) Liouville function znorder(a,n) multiplicative order of a mod n znprimroot(n) smallest primitive root chebyshev_theta(n) first Chebyshev function chebyshev_psi(n) second Chebyshev function hclassno(n) Hurwitz class number H(n) * 12 ramanujan_tau(n) Ramanujan's Tau function consecutive_integer_lcm(n) lcm(1 .. n) lucasu(P, Q, k) U_k for Lucas(P,Q) lucasv(P, Q, k) V_k for Lucas(P,Q) lucas_sequence(n, P, Q, k) (U_k,V_k,Q_k) for Lucas(P,Q) mod n bernfrac(n) Bernoulli number as (num,den) bernreal(n) Bernoulli number as BigFloat harmfrac(n) Harmonic number as (num,den) harmreal(n) Harmonic number as BigFloat stirling(n,m,[type]) Stirling numbers of 1st or 2nd type numtoperm(n,k) kth lexico permutation of n elems permtonum([a,b,...]) permutation number of given perm randperm(n,[k]) random permutation of n elems shuffle(...) random permutation of an array
NON-INTEGER MATH
ExponentialIntegral(x) Ei(x) LogarithmicIntegral(x) li(x) RiemannZeta(x) ζ(s)-1, real-valued Riemann Zeta RiemannR(x) Riemann's R function LambertW(k) Lambert W: solve for W in k = W exp(W) Pi([n]) The constant π (NV or n digits)
SUPPORT
prime_get_config gets hash ref of current settings prime_set_config(%hash) sets parameters prime_memfree frees any cached memory
COPYRIGHT
Copyright 2011-2018 by Dana Jacobsen <dana@acm.org>This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
| 2018-11-15 | perl v5.32.1 |
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