Number Theory Calculators

Euler Totient Calculator

Compute Euler's totient φ(n) — count of integers coprime to n

Congruence Equation Solver

Solve linear congruence equations ax≡b(mod m) with extended Euclidean algorithm

Prime Factorization Calculator

Decompose positive integers into prime factors with step-by-step trial division

Perfect & Amicable Number Checker

Determine if a number is perfect, amicable, or ordinary

Proper Divisor Calculator

List all proper divisors and compute their sum, count, and product

GCD & LCM Calculator

Compute the greatest common divisor and least common multiple using Euclidean algorithm

Carmichael Number Checker

Test if a number is a Carmichael (absolute pseudoprime) using Korselt criterion

Smith Number Checker

Check if sum of digits equals sum of digits of prime factors (Smith number)

Harshad Number Calculator

Test if a number is divisible by the sum of its digits (Niven number)

Pythagorean Triple Generator

Generate primitive and non-primitive Pythagorean triples using Euclid formula

Aliquot Sequence Calculator

Compute the sum-of-proper-divisors sequence and detect perfect/amicable/sociable numbers

Perfect Digital Invariant Calculator

Check if a number is an Armstrong (narcissistic) number

Happy Number Checker

Test if a number reaches 1 through digit square sum iteration (happy number)

Lucky Number Sieve Calculator

Generate and test lucky numbers using the Josephus elimination sieve process

Semiprime Number Finder

Check if a number is the product of exactly two primes (semiprime)

Deficient & Abundant Number Classifier

Classify numbers as deficient, perfect, or abundant based on proper divisor sum

Sociable Numbers Calculator

Find sociable cycles and detect perfect/amicable/sociable aliquot cycles

Mersenne Prime Checker

Test if a Mersenne number (2^n-1) is prime using Lucas-Lehmer test

Palindromic Prime Generator

Generate and test numbers that are both palindromic and prime

Beatty Sequence Calculator

Generate floor(n*alpha) Beatty sequence terms with Rayleigh complementary verification

Modular Square Root Calculator

Compute x satisfying x^2 = a (mod p) using Euler criterion and Tonelli-Shanks

Prime Number Checker

Test whether a positive integer is prime or composite with step-by-step divisibility checks

Coprime Calculator

Check whether two integers are relatively prime by computing gcd with the Euclidean algorithm

Modular Inverse Calculator

Find a inverse modulo m using the extended Euclidean algorithm and gcd condition

Chinese Remainder Theorem Calculator

Solve two modular congruences and combine them into one CRT solution

Divisibility Rules Calculator

Check common divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10, 11, and 12

Prime Counting Function Calculator

Calculate pi(n), the number of primes less than or equal to n, using a sieve method

Modular Exponentiation Calculator

Compute a^b mod m using repeated squaring with step-by-step modular arithmetic

Legendre Symbol Calculator

Evaluate the Legendre symbol and check quadratic residue status modulo an odd prime

Mobius Function Calculator

Calculate mu(n) from prime factorization, square-free status, and factor count

Divisor Sum Function Calculator

Calculate sigma(n), the sum of all positive divisors, using prime factorization

Twin Prime Checker

Check whether n and n+2 are both prime and form a twin prime pair

Prime Gap Calculator

Find nearby primes around n and calculate the local prime gap

Jacobi Symbol Calculator

Compute the Jacobi symbol for odd n using reciprocity and modular reductions

Multiplicative Order Calculator

Find the smallest k where a^k is congruent to 1 modulo n

Primitive Root Checker

Check whether a candidate is a primitive root modulo n by comparing order with phi(n)

Square-Free Number Checker

Check whether an integer is square-free using prime factorization

Radical of an Integer Calculator

Calculate rad(n), the product of distinct prime factors of an integer

Perfect Power Checker

Test whether an integer can be written as a^k with k greater than 1