Prime numbers: the atoms of maths

Everything around you is made from tiny building blocks called atoms. Whole numbers have building blocks too. They are called prime numbers, and every other whole number bigger than 1 can be built by multiplying them together.

Prime or composite?

A factor is a number that divides into another number with nothing left over. The factors of 12 are 1, 2, 3, 4, 6 and 12. A prime number has exactly two factors: 1 and itself. So 13 is prime, because only 1 × 13 makes it.

The sieve

More than 2,000 years ago, a Greek mathematician called Eratosthenes found a quick way to find primes. Write out the numbers to 100. Cross out every multiple of 2 (except 2), then of 3, then of 5, then of 7. The numbers left are all prime. There are 25 of them below 100.

Prime fingerprints

Split a number into factors, then split those factors again, until only primes are left. This is a factor tree. However you start, you always end with the same primes. For 36 it is 2 × 2 × 3 × 3, which we can write with powers as 2² × 3². Every number has its own prime fingerprint.

HCF and LCM

The highest common factor (HCF) is the biggest number that divides into two numbers. The HCF of 24 and 36 is 12, so 24 performers and 36 visitors can be split into groups of 12. The lowest common multiple (LCM) is the smallest number both divide into. The LCM of 8 and 12 is 24, so buying 3 packs of 8 sausages and 2 packs of 12 buns leaves nothing over.

Primes in nature and on the internet

Some cicadas in North America stay underground for 13 or 17 years, both primes. Scientists think this helps them avoid predators with shorter cycles. People also use primes to keep secrets: online banking depends on the fact that multiplying two huge primes is easy, but splitting the answer back into them is very, very hard. In 2024 a volunteer found the largest known prime. It has more than 41 million digits.