# NAME

ntheory - Number theory utilities

# SEE

See Math::Prime::Util for complete documentation.

# QUICK REFERENCE

Tags: :all to import almost all functions :rand to import rand, srand, irand, irand64

## PRIMALITY

``````  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_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
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
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
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
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 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
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,\\$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
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 2011-2017 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.