Free Number Theory Calculators

GCD & LCM Calculator | Up to 10 Numbers

Find the GCD and LCM of up to 10 integers using the Euclidean algorithm. Shows prime factorization, divisibility check, and full step-by-step working.

Prime Factorization Calculator

Find the prime factorization of any positive integer and check primality.

Factorial Calculator | n!, n!! & Derangements

Calculate standard factorial (n!), double factorial (n!!), and subfactorial (!n / derangements) with exact BigInt arithmetic, scientific notation, digit count, trailing zeros, Stirling approximation, and step-by-step working.

Fibonacci Calculator | F(n), Sequence & Check

Find any Fibonacci number F(n) with exact BigInt arithmetic up to F(10000), generate sequences with golden ratio convergence, and check whether any integer is a Fibonacci number using the algebraic 5n²±4 perfect-square test.

Prime Number Checker

Check if any number is prime using an optimized trial division algorithm. Also lists factors.

Perfect Number Checker

Check if a number is perfect, abundant, or deficient and find all its proper divisors.

Roman Numeral Converter

Convert between Arabic numbers and Roman numerals (I, V, X, L, C, D, M).

Number Sequence Calculator

Find the nth term of arithmetic, geometric, and other common sequences.

Digital Root Calculator

Calculate the digital root (iterated sum of digits), additive persistence, Harshad check, and multiplicative digital root for any integer up to 300 digits.

Modular Arithmetic Calculator | Modular Inverse, CRT & Power Mod

Perform all core modular arithmetic operations with full step-by-step working. Computes (a op b) mod n, finds modular inverses using the Extended Euclidean Algorithm, solves two-congruence systems via the Chinese Remainder Theorem, and computes modular exponentiation with repeated squaring.

Continued Fraction Calculator | Convergents & Best Rational Approximations

Convert any decimal or fraction to its continued fraction expansion [a₀; a₁, a₂, ...] with up to 12 terms. Computes the full convergent table showing the best rational approximations at each step, the approximation error, and evaluates any given continued fraction back to its decimal value.

Big Number Calculator | Arbitrary Precision Arithmetic

Perform exact arithmetic on integers of any size using JavaScript BigInt. Supports addition, subtraction, multiplication, exponentiation, factorial, GCD, LCM, and modular arithmetic with no floating-point rounding errors. Displays the number of digits in the result and checks whether the result is prime using the Miller-Rabin primality test.

Euler's Totient Calculator | φ(n), Coprimes & Euler's Theorem

Compute Euler's totient function φ(n) — the count of integers from 1 to n that are coprime with n. Shows the prime factorization method, lists all coprimes for n ≤ 500, demonstrates multiplicativity φ(mn) = φ(m)φ(n), and verifies Euler's theorem aᵠ⁽ⁿ⁾ ≡ 1 (mod n) for any chosen a.

Catalan Number Calculator | Sequence, Formula & Applications

Compute Catalan numbers Cₙ up to n = 30 using exact arithmetic. Shows the closed-form formula C(2n,n)/(n+1), the recursive formula, and a table of all values. Explains applications: number of valid bracket sequences, binary tree shapes, convex polygon triangulations, mountain ranges, and ballot paths.

Collatz Sequence Calculator | 3n+1 Conjecture, Stopping Time & Path

Explore the Collatz conjecture (3n+1 problem) for any positive integer. Computes the complete sequence path, stopping time, total steps, peak value, and odd-step count. Visualizes the trajectory and lets you compare stopping times across a range of starting values.

Chinese Remainder Theorem Calculator | Simultaneous Congruences Solver

Solve a system of simultaneous congruences x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂), … using the Chinese Remainder Theorem. Applies Garner's algorithm for pairwise-coprime moduli and shows step-by-step Bézout coefficient computation.