ntheory(3pm)

✨ NAME ✨

ntheory — Number theory utilities. It's the ntheory(3pm) page from the User Contributed Perl Documentation corner, and she's literally here to hand you every number theory tool in the group chat. 🎀💖

👀 SEE ALSO 👀

See Math::Prime::Util for complete documentation. That's the full lore, bestie. 📚✨

🌈 QUICK REFERENCE 🌈

Okay so this is the quick-reference cheat sheet. Keep up, no cap. 💅

🏷️ Tags 🏷️

• :all — to import all functions (other than NON-EXPORTED below). 🫶
• :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). 🐌 she said what she said
• is_ramanujan_prime(n) — is n a Ramanujan prime. 👑
• is_gaussian_prime(a,b) — is a+bi a Gaussian 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. 💖
• prime_powers([start,] end) — array ref of prime powers. ✨
• twin_primes([start,] end) — array ref of twin primes. 👯
• semi_primes([start,] end) — array ref of semiprimes. 🌗
• almost_primes(k, [start,] end) — array ref of k-almost-primes. 🧮
• omega_primes(k, [start,] end) — array ref of k-omega-primes. Ω
• 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. ⬅️
• next_prime_power(n) — next prime power > n. ➡️
• prev_prime_power(n) — previous prime power < 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 prime count. 〰️
• prime_power_count(n) — count of prime powers <= n. 🔢
• prime_power_count(start, end) — count of prime powers in range. 📏
• prime_power_count_lower(n) — fast lower bound for prime power count. ⬇️
• prime_power_count_upper(n) — fast upper bound for prime power count. ⬆️
• prime_power_count_approx(n) — fast approximate prime power count. 〰️
• 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. 〰️
• nth_prime_power(n) — the nth prime power (n=1 returns 2). 🥇
• nth_prime_power_lower(n) — fast lower bound for nth prime power. ⬇️
• nth_prime_power_upper(n) — fast upper bound for nth prime power. ⬆️
• nth_prime_power_approx(n) — fast approximate nth prime power. 〰️
• 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 approximate twin prime count. 〰️
• 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 semiprime count. 〰️
• nth_semiprime(n) — the nth semiprime. 🥇
• nth_semiprime_approx(n) — fast approximate nth semiprime. 〰️
• almost_prime_count(k,n) — count of k-almost-primes. 🧮
• almost_prime_count_approx(k,n) — fast approximate k-almost-prime count. 〰️
• almost_prime_count_lower(k,n) — fast k-almost-prime count lower bound. ⬇️
• almost_prime_count_upper(k,n) — fast k-almost-prime count upper bound. ⬆️
• nth_almost_prime(k,n) — the nth number with exactly k factors. 🥇
• nth_almost_prime_approx(k,n) — fast approximate nth k-almost prime. 〰️
• nth_almost_prime_lower(k,n) — fast nth k-almost prime lower bound. ⬇️
• nth_almost_prime_upper(k,n) — fast nth k-almost prime upper bound. ⬆️
• omega_prime_count(k,n) — count divisible by exactly k primes. Ω
• nth_omega_prime(k,n) — the nth number div by exactly k primes. 🥇
• 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. 🔄
• inverse_li_nv(x) — float 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. 💻

🔁 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. 🔁
• foralmostprimes {...} k,[beg,],end — loop over k-almost-primes in range. 🔁
• forfactored {...} [start,] end — loop with factors. 🔁
• forsquarefree {...} [start,] end — loop with factors of square-free n. 🔁
• forsquarefreeint {...} [start,] end — loop over 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([start]) — returns a simple prime iterator. 🔁
• prime_iterator_object([start]) — returns a prime iterator object. 🔁
• lastfor — stop iteration of for.... loop. 🛑

🎲 RANDOM NUMBERS 🎲

• irand() — random 32-bit integer. 🎲
• irand64() — random UV-bit integer (64 or 32). 🎲
• 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_safe_prime(n) — random safe 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 🌸

okay so bestie, this section is a whole vibe — we've got vector/list helpers and integer set helpers, all lined up and ready. every signature is here, no cap. 👑

• vecsum(@list) — gives you the integer sum of the list. ✨
• vecprod(@list) — same energy, but it's the integer product of the list. 💅
• vecmin(@list) — finds the minimum of a list of integers. lowkey essential.
• vecmax(@list) — finds the maximum of a list of integers. highkey iconic.
• vecuniq(@list) — removes duplicates from a list of integers. so clean ✨
• vecsingleton(@list) — removes all items that aren't unique, so only the one-of-a-kind ones stay. 💀
• vecfreq(@list) — returns a hash where each item maps to its count in the list. iykyk.
• vecsort(@list) — numerically sorts a list of integers. periodt.
• vecsorti(\@list) — sorts a list ref in place, numerically. no copy, just vibes. 💅
• vecextract(\@list, mask) — selects from the list based on a mask. it's giving filtered. 🎀
• vecequal(\@list1, \@list2) — compares two array refs for equality. are they twins? 👯
• vecreduce { ... } @list — does a reduce / left fold over the list with your block. ✨
• vecall { ... } @list — returns true if all elements make your block true. all or nothing.
• vecany { ... } @list — returns true if any element makes your block true. just one is enough.
• vecnone { ... } @list — returns true if none of the elements make your block true. zero vibes.
• vecnotall { ... } @list — returns true if not all elements make your block true. almost, bestie.
• vecfirst { ... } @list — returns the first value that makes your block evaluate true. 🥇
• vecfirstidx { ... } @list — returns the first index that makes your block evaluate true. where it at? 🔍
• vecmex(@list) — returns the least non-negative value not in the list. mex, but make it fashion. 💅
• vecpmex(@list) — returns the least positive value not in the list. pmex, periodt.
• vecsample(k,@list) — returns k random elements from the list. surprise, bestie! 🎲
• vecslide { ... } @list — calls your block for each pair in the list. sliding through like 🛝

---

• toset(...) — converts its arguments into an integer set (a unique sorted array ref). so organized. 🧺
• setinsert(\@A,$v) — inserts integer v into integer set A. welcome to the set. 🫶
• setinsert(\@A,\@B) — inserts all values from list B into integer set A. bulk add, no cap.
• setremove(\@A,$v) — removes integer v from integer set A. bye bestie. 👋
• setremove(\@A,\@B) — removes all values from list B from integer set A. mass removal. 💀
• setinvert(\@A,$v) — if v is in set A, removes it; otherwise adds it. flip flop. 🔄
• setinvert(\@A,\@B) — does that invert thing for every value in integer set B. invert the whole vibe. ✨
• setcontains(\@A,...) — checks whether all the listed values are in integer set A. are they all here? 👀
• setcontains(\@A,\@B) — checks whether integer set B is a subset of integer set A. tiny but mighty.
• setcontainsany(\@A,...) — checks whether any of the listed values are in integer set A. just one? valid.
• setcontainsany(\@A,\@B) — checks whether any value in B is in integer set A. overlap check. 🔍
• setbinop { ... } \@A[,\@B] — applies your operation block to all a,b pairs where a is from A and b is from B (B is optional). binary op energy. ⚡
• sumset(\@A[,\@B]) — applies a+b to all a,b pairs where a is from A and b is from B (B is optional). math but cute. ➕
• setunion(\@A,\@B) — union of two integer lists. together forever. 🫶
• setintersect(\@A,\@B) — intersection of two integer lists. the overlap. 💞
• setminus(\@A,\@B) — difference of two integer lists. what's left over. ✨
• setdelta(\@A,\@B) — symmetric difference of two integer lists. the exclusive or vibe. 🔀
• is_sidon_set(\@L) — checks if integer list L is a Sidon set. fancy. 👑
• is_sumfree_set(\@L) — checks if integer list L is a sum-free set. no sums allowed. 🚫
• set_is_disjoint(\@A,\@B) — checks if set B is disjoint from set A. zero overlap. 🚷
• set_is_equal(\@A,\@B) — checks if set B is equal to set A. same exact energy. 💯
• set_is_subset(\@A,\@B) — checks if set B is a subset of set A. smaller but included. 🎀
• set_is_proper_subset(\@A,\@B) — checks if set B is a proper subset of set A. subset but not equal. strict vibes.
• set_is_superset(\@A,\@B) — checks if set B is a superset of set A. bigger and contains all. 👑
• set_is_proper_superset(\@A,\@B) — checks if set B is a proper superset of set A. superset but not equal. extra. 💅
• set_is_proper_intersection(\@A,\@B) — checks if set B is a proper intersection of set A. intersection with rules. iykyk.

🧮 MATH 🧮

Okay bestie, buckle up 💖 this is the whole number-theory toolbox, every single function, no skips, no cap. Let's go down the list together 🌸

• todigits(n[,base[,len]]) — turns n into an array of its digits in whatever base you want
• todigitstring(n[,base[,len]]) — same energy but you get a string back instead of an array 💅
• fromdigits(\@d,[,base]) — takes an arrayref of digits and converts that base-digit vector back into a number
• fromdigits(str,[,base]) — takes a digit string and converts it back into a number ✨
• sumdigits(n) — adds up all the digits of n, and yes there's an optional base
• tozeckendorf(n) — converts n into Zeckendorf / Fibbinary form 🦋
• fromzeckendorf(str) — the glow-up in reverse: a Zeckendorf binary string becomes a number again
• is_odd(n) — returns 1 if n is odd, 0 otherwise
• is_even(n) — returns 1 if n is even, 0 otherwise
• is_divisible(n,d) — returns 1 if n is divisible by d
• is_congruent(n,c,d) — returns 1 if n is congruent to c mod d
• is_qr(a,n) — returns 1 if a is a quadratic residue mod n
• is_square(n) — returns 1 if n is a perfect square 🎀
• is_power(n) — returns k if n = c^k for some integer c
• is_power(n,k) — returns 1 if n = c^k for an integer c and that specific k
• is_power(n,k,\$root) — same as above but it also drops c into $root for you
• is_perfect_power(n) — returns 1 if n = c^k where c != 0 and k > 1
• is_prime_power(n) — returns k if n = p^k for a prime p and k > 0
• is_prime_power(n,\$p) — same deal but it also sets $p to p 💖
• is_square_free(n) — true when no prime factor shows up more than once (no repeated factors)
• is_powerfree(n[,k]) — is n free of any k-th powers?
• is_cyclic(n) — does n have only one group of order n?
• 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) — gives you v where v! % n == n-1 and n % v != 1
• is_semiprime(n) — does n have exactly 2 prime factors? 👀
• is_almost_prime(k,n) — does n have exactly k prime factors?
• is_omega_prime(k,n) — is n divisible by exactly k primes?
• is_chen_prime(n) — is n prime and n+2 prime or semiprime?
• is_polygonal(n,k) — is n a k-polygonal number?
• is_polygonal(n,k,\$root) — same, but it also sets $root for you ✨
• is_sum_of_squares(n[,k]) — is n a sum of k squares (default is 2)?
• is_congruent_number(n) — is n a congruent number?
• is_perfect_number(n) — is n equal to the sum of its proper divisors? iconic behavior tbh
• is_fundamental(d) — is d a fundamental discriminant?
• is_totient(n) — is n = euler_phi(x) for some x?
• is_lucky(n) — is n a lucky number? 🍀
• is_happy(n) — if n is a happy number it returns the height 😊
• is_happy(n,base,exponent) — same but for S_b_e happy numbers, returns the height
• is_smooth(n,k) — is n a k-smooth number?
• is_rough(n,k) — is n a k-rough number?
• is_powerful(n[,k]) — is n a k-powerful number?
• is_practical(n) — is n a practical number?
• is_delicate_prime(n) — is n a digitally delicate prime? 🙈
• powint(a,b) — signed integer a^b
• mulint(a,b) — signed integer a * b
• addint(a,b) — signed integer a + b
• subint(a,b) — signed integer a - b
• add1int(n) — signed integer n + 1
• sub1int(n) — signed integer n - 1
• divint(a,b) — signed integer a / b (floor)
• modint(a,b) — signed integer a % b (floor)
• cdivint(a,b) — signed integer a / b (ceilint)
• divrem(a,b) — returns (quot,rem) of a/b (Euclidian)
• fdivrem(a,b) — returns (quot,rem) of a/b (floored)
• cdivrem(a,b) — returns (quot,rem) of a/b (ceiling)
• tdivrem(a,b) — returns (quot,rem) of a/b (truncated)
• lshiftint(n,k) — left shift n by k bits
• rshiftint(n,k) — right shift n by k bits (truncate)
• rashiftint(n,k) — right shift n by k bits (floor)
• absint(n) — integer absolute value
• negint(n) — integer negation
• cmpint(a,b) — integer comparison, works like <=>
• signint(n) — integer sign as -1, 0, or 1
• sqrtint(n) — integer square root
• rootint(n,k) — integer k-th root
• rootint(n,k,\$rk) — same but it also sets $rk to r^k 🫶
• logint(n,b) — integer logarithm
• logint(n,b,\$be) — same but it also sets $be to b^e
• gcd(@list) — greatest common divisor
• lcm(@list) — least common multiple
• gcdext(x,y) — returns (u,v,d) where u*x+v*y=d
• chinese([a,mod1],[b,mod2],...) — CRT that returns the remainder
• chinese2([a,mod1],[b,mod2],...) — CRT that returns (remainder,LCM)
• frobenius_number(@list) — the Frobenius Number of a set
• primorial(n) — product of all primes below n
• pn_primorial(n) — product of the first n primes
• factorial(n) — product of the first n integers, aka n!
• factorialmod(n,m) — factorial mod m
• subfactorial(n) — count of derangements of n objects
• binomial(n,k) — binomial coefficient
• binomialmod(n,k,m) — binomial(n,k) mod m
• falling_factorial(x,n) — falling factorial
• rising_factorial(x,n) — rising factorial
• partitions(n) — number of integer partitions
• valuation(n,k) — how many times n is divisible by k
• hammingweight(n) — population count, aka the number of binary 1s
• kronecker(a,b) — Kronecker (Jacobi) symbol
• negmod(a,n) — -a mod n
• addmod(a,b,n) — a + b mod n
• submod(a,b,n) — a - b mod n
• mulmod(a,b,n) — a * b mod n
• muladdmod(a,b,c,n) — a * b + c mod n
• mulsubmod(a,b,c,n) — a * b - c 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
• rootmod(a,k,n) — modular k-th root
• allsqrtmod(a,n) — list of all the modular square roots
• allrootmod(a,k,n) — list of all the modular k-th roots
• cornacchia(d,n) — finds x,y for x^2 + d * y^2 = n
• prime_bigomega(n) — number of prime factors
• prime_omega(n) — number of distinct prime factors
• moebius(n) — Moebius function of n
• moebius(beg, end) — list of Moebius values over a range
• mertens(n) — sum of Moebius from 1 to n
• euler_phi(n) — Euler totient of n
• euler_phi(beg, end) — Euler totient over a range
• inverse_totient(n) — the image of Euler totient
• jordan_totient(k,n) — Jordan's totient
• sumtotient(n) — sum of Euler totient from 1 to n
• carmichael_lambda(n) — Carmichael's Lambda function
• ramanujan_sum(k,n) — Ramanujan's sum
• exp_mangoldt(n) — exponential of the Mangoldt function
• liouville(n) — Liouville function
• sumliouville(n) — sum of Liouville from 1 to n
• znorder(a,n) — multiplicative order of a mod n
• znprimroot(n) — smallest primitive root
• znlog(a, g, p) — solves for k in a = g^k mod p
• qnr(n) — least quadratic non-residue
• 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)
• lucasuv(P, Q, k) — (U_k,V_k) for Lucas(P,Q)
• lucasumod(P, Q, k, n) — U_k for Lucas(P,Q) mod n
• lucasvmod(P, Q, k, n) — V_k for Lucas(P,Q) mod n
• lucasuvmod(P, Q, k, n) — (U_k,V_k,Q^k) for Lucas(P,Q) mod n
• lucas_sequence(n,P,Q,k) — deprecated, please use lucasuvmod instead bestie, fr fr
• pisano_period(n) — the period of Fibonacci numbers mod n
• bernfrac(n) — Bernoulli number as (num,den)
• bernreal(n) — Bernoulli number as a BigFloat
• harmfrac(n) — Harmonic number as (num,den)
• harmreal(n) — Harmonic number as a BigFloat
• stirling(n,m,[type]) — Stirling numbers, 1st or 2nd type, your pick ✨
• fubini(n) — Fubini (Ordered Bell) number
• numtoperm(n,k) — the k-th lexico permutation of n elements
• permtonum([a,b,...]) — permutation number of the permutation you feed it
• randperm(n,[k]) — random permutation of n elements 🎲
• shuffle(...) — random permutation of an array
• lucky_numbers(n) — array ref of the lucky sieve up to n
• lucky_count(n) — count of lucky numbers <= n
• lucky_count(start, end) — count of lucky numbers in a range
• lucky_count_lower(n) — fast lower bound for the lucky count
• lucky_count_upper(n) — fast upper bound for the lucky count
• lucky_count_approx(n) — fast approximate lucky count
• nth_lucky(n) — the n-th entry in the lucky sieve
• nth_lucky_lower(n) — fast lower bound for the n-th lucky number
• nth_lucky_upper(n) — fast upper bound for the n-th lucky number
• nth_lucky_approx(n) — fast approximate n-th lucky number
• minimal_goldbach_pair(n) — the least prime p where n-p is also prime
• goldbach_pair_count(n) — how many prime pairs sum to n
• goldbach_pairs(n) — array of every p where p and n-p are both prime
• powerful_numbers([lo,]hi[,k]) — array ref of k-powerful numbers from lo to hi
• powerful_count(n[,k]) — count of k-powerful numbers <= n
• sumpowerful(n[,k]) — sum of k-powerful numbers <= n
• nth_powerful(n[,k]) — the n-th k-powerful number
• next_perfect_power(n) — the next perfect power greater than n 👑

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• prev_perfect_power(n) — the perfect power that comes right before < n 🍓
• perfect_power_count(n) — how many perfect powers are <= n
• perfect_power_count(start, end) — how many live inside that range
• perfect_power_count_lower(n) — fast lower bound for the perfect power count
• perfect_power_count_upper(n) — fast upper bound for the perfect power count
• perfect_power_count_approx(n) — fast approximate perfect power count
• nth_perfect_power(n) — the *n*th perfect power, periodt ✨
• nth_perfect_power_lower(n) — fast lower bound for the nth perfect power
• nth_perfect_power_upper(n) — fast upper bound for the nth perfect power
• nth_perfect_power_approx(n) — fast approximate nth perfect power
• next_chen_prime(n) — the next Chen prime > n
• smooth_count(n,k) — count of k-smooth numbers <= n
• rough_count(n,k) — count of k-rough numbers <= n
• powerfree_count(n[,k]) — count of k-powerfree numbers <= n
• nth_powerfree(n[,k]) — the nth k-powerfree number, slay
• powerfree_sum(n[,k]) — sum of the k-powerfree numbers <= n
• powerfree_part(n[,k]) — strips the excess powers so n comes out k-free
• powerfree_part_sum(n[,k]) — sum of the k-powerfree parts for 1 to n
• squarefree_kernel(n) — the integer radical of |n|
• powersum(n,k) — sum of the kth powers from 1 to n 🌸

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✨ RATIONALS ✨

• contfrac(n,d) — the continued fraction for n/d, as a list
• from_contfrac(@A) — gives you back the (p,q) rational from a cfrac list
• next_calkin_wilf(n,d) — next breadth-first CW rational
• next_stern_brocot(n,d) — next breadth-first SB rational
• calkin_wilf_n(n,d) — the index of a breadth-first CW rational
• stern_brocot_n(n,d) — the index of a breadth-first SB rational
• nth_calkin_wilf(n) — the CW rational sitting at breadth-first index n
• nth_stern_brocot(n) — the SB rational sitting at breadth-first index n
• nth_stern_diatomic(n) — Stern's Diatomic series; fusc(n) 💜
• farey(n) — the Farey sequence of order n, listed out
• farey(n,k) — the k'th entry of the Farey sequence of order n
• next_farey(n,[p,q]) — the next order-n rational after p/q
• farey_rank(n,[p,q]) — how many things in F_n are less than p/q

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🦋 NON-INTEGER MATH 🦋

• ExponentialIntegral(x) — Ei(x) 👑
• LogarithmicIntegral(x) — li(x)
• RiemannZeta(x) — ζ(s)-1, the 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, either as an NV or as n digits, your call 💅

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🌸 SUPPORT 🌸

• prime_get_config — hands you a hash ref of the current settings
• prime_set_config(%hash) — sets those parameters
• prime_memfree — frees up any cached memory, bye girl 🫶

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💅 ADDITIONAL NON-EXPORTED 💅

• trial_factor(n[,limit]) — factor using only trial division
• fermat_factor(n) — factor using only Fermat's method
• holf_factor(n[,rounds]) — factor using only Hart's OLF
• lehman_factor(n) — factor using only Lehman (limited size, bestie, it has a ceiling)
• squfof_factor(n[,rounds]) — factor using only SQUFOF
• prho_factor(n[,rounds]) — factor using only Pollard's Rho
• pbrent_factor(n[,rounds]) — factor using only Brent/Pollard Rho
• pminus1_factor(n[,B1[,B2]]) — factor using only P-1
• pplus1_factor(n[,B]) — factor using only P+1
• cheb_factor(n[,B1[,initx]]) — factor using only Chebyshev
• ecm_factor(n[,B1[,B2[,curves]]]) — factor using only ECM 🔥
• _uvbits — size of UV in bits
• _uvsize — size of UV in bytes
• _ivsize — size of IV in bytes
• _nvsize — size of NV in bytes
• _nvmantbits — bits stored in the NV mantissa
• _nvmantdigits — count of whole decimal digits in an NV

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🧚 ADDITIONAL NON-EXPORTED C ONLY 🧚

(yeah — these ones are C only, no Perl for you 🙈)

• _segment_pi(n) — prime count using only sieving
• _legendre_pi(n) — prime count with the Legendre method
• _meissel_pi(n) — prime count with the Meissel method
• _lehmer_pi(n) — prime count with the Lehmer method
• _LMO_pi(n) — prime count with the LMO method
• _LMOS_pi(n) — prime count with the extended LMO method

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👑 COPYRIGHT 👑

Copyright 2011-2026 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.

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and that's the whole page, bestie, we made it 💖 this is ntheory(3pm) as it ships with perl v5.42.3, dated 2026-08-20 🌈 go forth and factor something iconic ✨🎀