factorial
n! -- the product of every whole number from 1 to n.
n(number) -- a non-negative whole number
Number Theory
nCr
The number of ways to choose r items from n, order not mattering (n choose r).
n(number) -- total itemsr(number) -- items chosen
nPr
The number of ways to arrange r items from n, order mattering (n permute r).
n(number) -- total itemsr(number) -- items arranged
catalan
The nth Catalan number -- counts things like balanced bracket sequences and binary tree shapes.
n(number) -- index, starting at 0
fibonacci
The nth Fibonacci number (0, 1, 1, 2, 3, 5, 8, ...).
n(number) -- a non-negative index; fibonacci(0) = 0
gcd
The greatest common divisor of a and b. Works at any size (past 10^38 through big integers).
a(number) --b(number) --
lcm
The least common multiple of a and b. Works at any size (past 10^38 through big integers).
a(number) --b(number) --
divides
True if a evenly divides b (b is a multiple of a).
a(number) -- the divisorb(number) -- the number being divided
quotient
The quotient q of the division algorithm a = b*q + r with 0 <= r < |b|, exact at any size. For a positive b it is floor(a/b); quotient(-17, 5) is -4 and quotient(17, -5) is -3.
a(integer) --b(integer) -- not 0
remainder
The remainder r of the division algorithm a = b*q + r with 0 <= r < |b|: never negative, exact at any size. The same as a % b when b is positive; remainder(-17, 5) is 3 and remainder(17, -5) is 2 (17 % -5 is -3).
a(integer) --b(integer) -- not 0
divmod
Quotient and remainder together, as {quotient, remainder}, from the division algorithm a = b*q + r with 0 <= r < |b|. Read them with r.quotient and r.remainder.
a(integer) --b(integer) -- not 0
sumDivisors
The sum of all positive divisors of n, including 1 and n itself.
n(number) --
countDivisors
How many positive divisors n has, including 1 and n itself.
n(number) --
mulmod
(a * b) mod m, after first reducing a and b modulo m so their full product is never formed. Works at any size (past 10^38 through big integers).
a(number) --b(number) --m(number) -- the modulus
modPow
base^exp mod m, computed efficiently (fast exponentiation) without ever forming the full power. Works at any size (past 10^38 through big integers).
base(number) --exp(number) --m(number) -- the modulus
modInv
The modular inverse of a mod m -- the value x where (a*x) mod m == 1. Returns explanatory text ('no inverse exists') when a and m share a common factor.
a(number) --m(number) -- the modulus
solveCongruence
Finds one solution to the linear congruence a*x = b (mod m). Returns explanatory text ('no solution exists') when none exists.
a(number) --b(number) --m(number) -- the modulus
fermatCheck
Checks whether a^(p-1) = 1 (mod p). A false result proves p is composite; a true result does not prove p is prime.
a(number) -- the base to test withp(number) -- the candidate prime
isPrime
True if n is a prime number. Works at any size (past 10^38 through big integers).
n(number) --
nextPrime
The smallest prime strictly greater than n. Works at any size (past 10^38 through big integers).
n(number) --
triangular
The nth triangular number, 1 + 2 + ... + n. Works at any size (past 10^38 through big integers).
n(number) --
isTriangular
True if n is a triangular number (1, 3, 6, 10, 15, ...).
n(number) --
piNaive
pi(n) -- the count of primes less than or equal to n, computed by direct trial (simple, fine for small n).
n(number) --
piSieve
pi(n) -- the count of primes less than or equal to n, computed via a sieve (fast for larger n).
n(number) --
nthPrime
The nth prime number (nthPrime(1) = 2, nthPrime(2) = 3, ...).
n(number) -- 1-indexed position
isSquareFree
True if no prime's square divides n -- n has no repeated prime factor.
n(number) --
isPerfectPower
True if positive integer n can be written as m^k for integers m and k >= 2 (e.g. 8 = 2^3, 9 = 3^2).
n(number) --
extendedGcd
x, y and gcd(a, b) with a*x + b*y = gcd(a, b) (extended Euclidean algorithm).
a(integer) --b(integer) --
twinPrimes
All twin prime pairs (p, p + 2) up to n.
n(integer) --
cousinPrimes
All cousin prime pairs (p, p + 4), both prime, up to n.
n(integer) --
sexyPrimes
All sexy prime pairs (p, p + 6), both prime, up to n.
n(integer) --
primePairs
All pairs (p, p + d) with both prime, up to n.
d(integer) --n(integer) --
consecutivePrimePairs
Consecutive primes exactly gap apart, up to n (prime gaps).
gap(integer) --n(integer) --
maxPrimeGap
The largest gap between consecutive primes up to n.
n(integer) --
isMersennePrime
True if 2^p - 1 is prime (Lucas-Lehmer).
p(integer) --
primeWithDigits
The smallest prime with the given number of digits (10^(d-1) rounded up to a prime).
digits(integer) --
digitCount
The number of decimal digits of a whole number of any size.
n(integer) --
toBase
A whole number written in base 2 to 36, as text.
n(integer) --base(integer) -- 2 to 36
fromBase
The number a base-2..36 string spells.
text(string) -- quoted, e.g. "ff"base(integer) -- 2 to 36
toHex
A whole number in hexadecimal.
n(integer) --
fromHex
The number a hexadecimal string spells.
text(string) -- quoted, e.g. "ff"