Prime and Composite Numbers

The primes up to 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47, which is 15 in all. 2 is the only even prime. Every other even number has 2 as a factor.

Try dividing by primes up to the square root of the number. To test 97: √97 is just under 10, so test 2, 3, 5 and 7. None divides 97, so it is prime. To test 91: 91 ÷ 7 = 13, so it is composite. The prime number checker does this instantly, even for very large values.

To list primes up to 50: write the numbers 2 to 50, circle 2 and cross out its multiples, circle the next uncrossed number (3) and cross out its multiples, and repeat. The circled numbers are the primes. You only need to sieve with primes up to √50 ≈ 7.

Every composite number is a product of primes in exactly one way. 84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7. Try any number in the prime factorization calculator.

There are infinitely many primes (Euclid). Pairs like 11 and 13 are called twin primes. Goldbach’s conjecture, that every even number above 2 is a sum of two primes, has been checked for huge numbers but is still unproven. The largest known primes have millions of digits and are found by computers.