Try arranging 12 buttons into a rectangle: you can make 2 rows of 6, or 3 rows of 4. Now try 7 buttons — you can only make one long row, no matter how you push them around. Numbers like 7 that can't be tidied into a fuller rectangle are called primes, and they're the building blocks of every other number. Run the sieve in the simulator and watch the primes light up, then pick a number and see it split into its prime pieces.
Most people think a number could be built from primes in several different ways. In fact every whole number above 1 factors into exactly one set of primes, no matter how you start splitting it, which is precisely why primes are called the atoms of multiplication.
What's actually happening
Most numbers are sociable: 12 can be cut into 2 sixes, 3 fours, or 4 threes, so you can lay 12 tiles into a proper rectangle several different ways. A prime number flatly refuses. Lay out 7 tiles, or 13, or 29, and the only rectangle you can make is a single straight row, one tile wide. That stubbornness is the whole definition — a prime has no whole-number divisors except 1 and itself.
The reason mathematicians care is that primes are where factoring bottoms out. Take any number and keep splitting it: 60 becomes 6 times 10, which becomes 2 times 3 times 2 times 5. You cannot break those last pieces any further, and you always land on the same set of primes no matter which way you started splitting. This is the fundamental theorem of arithmetic, and it means every number has one and only one prime recipe. Primes really are the atoms of multiplication, and every other number is a unique molecule built from them.
The strangest part is that the primes never run out and never fall into a pattern. Euclid proved, with a short and beautiful argument, that there can be no largest prime. Yet they thin out unpredictably as you climb, clumping and gapping with no formula that names the next one. That mix of total reliability and total unpredictability is exactly why primes guard your bank details: multiplying two huge primes together is easy, but pulling the product back apart into those primes is so hard that the world's online security leans on it.
Every number has one and only one prime recipe, which makes primes the unbreakable building blocks of all arithmetic.
- 1Grab a pile of coins or buttons and pick a number, say 15. Try to lay them into a full rectangle with more than one row.
- 2If you can (15 makes 3 rows of 5), it is not prime. If the only rectangle is a single row, like 11 or 17, you have found a prime.
- 3Work your way up from 2 to 20 and write down every number that only makes a single row. You have just rebuilt the start of the prime list by hand.
Common questions
Because every whole number above 1 factors into primes, and always into the same set no matter how you start splitting it. The primes are where factoring bottoms out, so each number is a unique molecule built from them.
Yes. Euclid proved around 300 BC, with a short and elegant argument, that there can be no largest prime. They keep going forever, though they thin out unpredictably with no formula naming the next one.
Multiplying two huge primes together is easy, but pulling the product back apart into those primes is extremely hard. RSA encryption relies on that gap to lock messages and bank details.