Number Theory Calculator - Advanced Number Properties & Operations

Prime Number Testing

Enter Number:

Prime Number Properties

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This calculator uses the Miller-Rabin primality test for large numbers.

Prime Factorization

Enter Number:

Fundamental Theorem of Arithmetic

Every integer greater than 1 can be uniquely represented as a product of prime numbers (up to the order of factors).

Divisor Functions

Enter Number:

Divisor Functions

τ(n) = number of divisors, σ(n) = sum of divisors. Perfect numbers have σ(n) = 2n.

Euler's Totient Function

Enter Number:

φ(n) = n × ∏(1 - 1/p) for all prime factors p of n

Euler's Totient Function

φ(n) counts the positive integers up to n that are relatively prime to n. Important in number theory and cryptography.

Linear Congruences

Coefficient a:

Constant b:

Modulus m:

ax ≡ b (mod m) has solutions iff gcd(a,m) | b

Linear Diophantine Equations

Coefficient a:

Coefficient b:

Constant c:

ax + by = c has integer solutions iff gcd(a,b) | c

🔢 Number Theory Calculator

Prime Number Testing

Prime Number Properties

Prime Factorization

Fundamental Theorem of Arithmetic

Divisor Functions

Divisor Functions

Euler's Totient Function

Euler's Totient Function

Linear Congruences

Linear Diophantine Equations