Universal Calculator
TAIIR

Prime Number Calculator

Check if a number is prime, find primes in any range, get complete prime factorization, and discover the nth prime number.

Quick Examples

Recent

No recent calculations

Prime Number Facts

  • 2 is the only even prime.
  • There are infinitely many primes.
  • Largest known prime has millions of digits.
  • Prime factorization is unique.

Mastering Prime Numbers: The Complete Guide

Prime numbers are the atoms of arithmetic—every integer greater than 1 can be uniquely expressed as a product of primes. This Professional Prime Number Calculator provides four essential tools: checking primality, performing prime factorization, listing primes in any range, and finding the nth prime. Whether you're exploring number theory, preparing for competitions, or need reliable factorization, this tool delivers instant, accurate results.

What Is a Prime Number?

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Notice that 2 is the only even prime—every even number greater than 2 is divisible by 2 and thus composite.

💡 Fundamental Theorem of Arithmetic

Every integer greater than 1 can be factored uniquely into prime numbers, up to the order of the factors. For example, 360 = 2³ × 3² × 5. This uniqueness is the cornerstone of number theory.

Checking Primality

To test if a number n is prime, it suffices to check divisibility by all primes up to √n. Our calculator uses an optimized trial division with 6k±1 optimization, efficiently handling numbers up to 10¹². For larger numbers, probabilistic tests like Miller–Rabin are used in advanced settings.

Prime Factorization

Factorization decomposes a composite number into a product of primes. This is crucial for computing GCD/LCM, simplifying radicals, and is the basis of RSA encryption—the difficulty of factoring large semiprimes ensures security.

Finding Primes in a Range

Using the Sieve of Eratosthenes, our calculator efficiently generates all primes up to a specified limit. This ancient algorithm marks multiples of each prime, leaving the unmarked numbers as primes. It's perfect for exploring prime distribution and gaps.

The Nth Prime

The nth prime function answers questions like "What is the 100th prime?" (it's 541). While no simple formula exists, our calculator uses an optimized sieve and primality test to quickly find primes up to the 100,000th.

Frequently Asked Questions

Why isn't 1 prime?

By definition, a prime must have exactly two distinct positive divisors. 1 has only one divisor (itself), so it's neither prime nor composite.

What is the largest known prime?

As of 2025, the largest known prime is 2⁸²⁵⁸⁹⁹³³−1, a Mersenne prime with over 24 million digits. Our calculator works with numbers up to 10¹² for practical use.

How are primes used in cryptography?

RSA encryption multiplies two large primes to create a public key. Decrypting requires factoring that product, which is computationally infeasible for sufficiently large primes.

What is the Sieve of Eratosthenes?

An ancient algorithm for finding all primes up to a limit. It iteratively marks multiples of each prime starting from 2, leaving unmarked numbers as primes.

Related Tools

Electricity CalculatorCapital Gains Tax CalculatorCalorie CalculatorBmr CalculatorBmi CalculatorArmy Body Fat Calculator