Number Theory Lab

Calculators and algorithms for discrete mathematics.

Parameters

Integers (comma-separated)

Integer

a

b

m (modulus)

Integer n

a

b

Integer n

GCD —

LCM —

Euclidean Algorithm Steps

Canonical Form

Divisors Count —

Sum of Divisors —

Number Type —

Factor Tree Steps

(a + b) mod m —

(a − b) mod m —

(a × b) mod m —

aᵇ mod m —

Modular Inverse a⁻¹ mod m —

a mod m (standard) —

φ(n) —

Coprime pairs with n —

Bézout\'s Identity

Algorithm Steps

— —

Miller-Rabin —

Next Prime —

Twin Prime Partner —

Mersenne? (if 2ⁿ−1) —

Number Theory Laboratory Handbook

Mathematical Principles & Theorems

Investigates core theorems in arithmetic and algebraic number theory: (1) Euclidean algorithm: \(\gcd(a, b) = \gcd(b, a \bmod b)\); (2) Extended Euclidean algorithm computes Bézout coefficients \(x, y\) such that \(ax + by = \gcd(a, b)\); (3) Modular multiplicative inverse \(a^{-1} \pmod m\) exists iff \(\gcd(a, m) = 1\); (4) Euler's totient \(\phi(n) = n\prod_{p|n}(1-1/p)\) and Euler's Theorem \(a^{\phi(n)} \equiv 1 \pmod n\); (5) Chinese Remainder Theorem (CRT) unique solution modulo \(\prod m_i\).

Operating Instructions

Enter integer operands \(a, b\) and modulus \(m\) in the parameter inputs.
Select number theory module: GCD / Extended Euclidean, Modular Inverse, Euler Totient, or CRT Solver.
Click Compute to obtain exact integer solutions and step-by-step Euclidean division lattices.
Review prime power factorizations and modular congruence proofs.