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. Numbers with more than two factors are called composite numbers.

Examples of Prime Numbers

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97...

A number n is prime if it has exactly two factors: 1 and n itself.
2 is the only even prime number.

How to Check if a Number is Prime

If n ≤ 1, it is not prime.
If n = 2 or n = 3, it is prime.
Check divisibility by all numbers from 2 to √n.
If divisible by any, it is composite. Otherwise, it is prime.

Primes up to 100

There are 25 prime numbers between 1 and 100. Memorising the smaller ones is genuinely useful for mental math:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Sieve of Eratosthenes

An ancient algorithm (c. 240 BC) to find all primes up to n efficiently:

List all integers 2 to n
Start with p=2. Mark all multiples of 2 (4, 6, 8...) as composite
Move to next unmarked number (3). Mark all multiples of 3 as composite
Continue until p > √n
Remaining unmarked numbers are all prime

Prime Number Theorems

Fact	Details
There are infinitely many primes	Proved by Euclid c. 300 BC
2 is the only even prime	All other even numbers divisible by 2
Every integer >1 has a unique prime factorization	Fundamental Theorem of Arithmetic
Largest known prime (2024)	2¹³⁶,²⁷⁹,⁸⁴¹−1 (Mersenne prime, ~41 million digits)

Real-World Applications

Cryptography: RSA encryption (used in HTTPS/banking) relies on the difficulty of factoring large numbers into primes. A typical RSA key uses two 150-digit primes.
Hash functions: Prime numbers are used in hash tables and checksums for optimal distribution
Cicadas: Some cicada species emerge every 13 or 17 years (prime numbers) — an evolutionary strategy to avoid predator cycles
Number theory: Foundation of modular arithmetic, congruences, and Diophantine equations

Prime Number Checker