Number theory

All integer work uses exact arbitrary-size integers (BigInt). Inputs accept decimal, 0x, 0b, and 0o literals with _ separators.

Primes & factors: Miller–Rabin primality (deterministic below 3.3 × 10²⁴, probabilistic above), factorisation by trial division and Pollard’s rho, τ, σ, φ, μ, ω, and Ω, all divisors, the neighbouring primes, and a sieve listing primes up to 20 million.
GCD & modular: gcd and lcm with Bézout coefficients, modular exponentiation and inverse, and solving a·x ≡ b (mod m). The Chinese remainder theorem accepts one congruence per line (2 mod 3) and handles moduli that are not coprime.
Bases & fractions: any base from 2 to 36; fractions to decimals with the repeating part in brackets (1/7 = 0.(142857)) and back; continued fractions and best rational approximations (π → 22/7, 355/113…).
Big numbers: exact n!, C(n, r), P(n, r), Fibonacci numbers by fast doubling, 2ⁿ, and Catalan numbers.
Roman numerals: both directions, 1–3999.

Factorising numbers whose two largest prime factors are both beyond about 20 digits can exceed the four-second limit.

Share link (under the title) copies a URL that reopens this tool exactly as you have it — every tab’s inputs, with the current tab selected. Opening such a link replaces the inputs this browser had saved for the tool.

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Companion to chapters 1, 2 and 23 (numbers, fractions and decimals, number theory) of Mathematics, Reconnected. Each input box has a short parameter, and the tab comes from the # part of the address, for example /tools/numbers/?a=1071&b=462#gcd or ?frac=1/7#bases. Share link writes the same parameters.

Tab Parameter Meaning Example

Tab	Parameter	Meaning	Example
`#factor`	`n`	integer to test and factorise	`n=391`
`#factor`	`sieve`	list the primes up to this	`sieve=400`
`#gcd`	`a`, `b`	gcd, lcm and Bézout coefficients of a and b	`a=1071&b=462`
`#gcd`	`m`, `e`	modulus and exponent: a mod m, aᵉ mod m, a⁻¹ mod m, and a·x ≡ b	`a=7&e=13&m=11`
`#gcd`	`crt`	congruences `a mod m`, separated by `;`	`crt=2 mod 3;3 mod 5`
`#bases`	`num`, `base`	a number written in base `base` (2–36), shown in other bases	`num=ff&base=16`
`#bases`	`frac`	a fraction to a repeating decimal, or a repeating decimal back (`0.1(6)`)	`frac=1/7`
`#bases`	`x`	best rational approximations of a real number (`pi`, `e`, `√2`, `phi` or a decimal)	`x=pi`
`#combi`	`big`, `r`	n!, C(n, r), P(n, r), Fibonacci, 2ⁿ and Catalan for n = `big`	`big=100&r=7`
`#roman`	`roman`	a number or Roman numeral	`roman=MMXXVI`
	`example`	one of the presets below (it also picks the tab); other parameters override it	`example=ch23-jugs-3-5`

#factor

n

integer to test and factorise

n=391

#factor

sieve

list the primes up to this

sieve=400

#gcd

a, b

gcd, lcm and Bézout coefficients of a and b

a=1071&b=462

#gcd

m, e

modulus and exponent: a mod m, aᵉ mod m, a⁻¹ mod m, and a·x ≡ b

a=7&e=13&m=11

#gcd

crt

congruences a mod m, separated by ;

crt=2 mod 3;3 mod 5

#bases

num, base

a number written in base base (2–36), shown in other bases

num=ff&base=16

#bases

frac

a fraction to a repeating decimal, or a repeating decimal back (0.1(6))

frac=1/7

#bases

x

best rational approximations of a real number (pi, e, √2, phi or a decimal)

x=pi

#combi

big, r

n!, C(n, r), P(n, r), Fibonacci, 2ⁿ and Catalan for n = big

big=100&r=7

#roman

roman

a number or Roman numeral

roman=MMXXVI

example

one of the presets below (it also picks the tab); other parameters override it

example=ch23-jugs-3-5

Presets:

ch01-base-explorer — 255 in bases 2, 8, 10, 16 and more
ch01-sqrt2-approximations — Best fractions for √2: 7/5, 17/12, 41/29, …
ch01-factor-explorer — Factorise 360, with its divisors
ch02-repeating-1-7 — 1/7 = 0.(142857), period 6
ch02-repeating-1-17 — 1/17 has the maximum period, 16
ch02-recurring-36 — 0.(36) back to a fraction: 4/11
ch02-recurring-16 — 0.1(6) back to a fraction: 1/6
ch03-square-and-multiply — 3^1000000 mod 1,000,000,007 by square-and-multiply
ch23-is-391-prime — Is 391 prime? 17 × 23
ch23-sieve-30 — The primes up to 30
ch23-sieve-400 — The primes up to 400 (the 20 × 20 sieve)
ch23-gcd-1071-462 — gcd(1071, 462) = 21 = 7·462 − 3·1071
ch23-lcm-252-198 — gcd and lcm of 252 and 198
ch23-jugs-3-5 — Jugs of 3 and 5: gcd 1, so 4 gallons is reachable (Bézout)
ch23-inverse-7-mod-26 — 7⁻¹ mod 26 = 15
ch23-fermat-3-100-mod-7 — 3¹⁰⁰ mod 7 = 4
ch23-modpow-7-13-mod-11 — 7¹³ mod 11 = 2 by square-and-multiply
ch23-diffie-hellman — Diffie–Hellman: 5⁶ mod 23 = 8
ch23-rsa-private-key — RSA with p = 5, q = 11: d = 3⁻¹ mod 40 = 27
ch23-rsa-decrypt — RSA: decrypt 13²⁷ mod 55 = 7
ch23-crt — Chinese remainder theorem: x ≡ 2 (mod 3), 3 (mod 5), 2 (mod 7)

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