Every textbook on number theory will begin with a treatise on prime numbers; every treatise on prime numbers will begin by emphasising their importance as building blocks or atoms of our number system: every integer can be expressed as a product of prime numbers in one way *and one way only*. Six is two times three and there is no other way to decompose it.^{1} Euclid proved this over two thousand years ago and it is so fundamental (hence the name *fundamental theorem of arithmetic*) to our thinking about numbers that we take it for granted. It is not!

There are numbers that behave very much like the integers but have a different structure. One rather simple example are the Gaussian integers (usually denoted ) which look just like complex numbers , except that and are restricted to integer values. They live in the complex plane, but exclusively on a discrete grid amongst their continuous cousins.

Continue reading Primes from a Different World

Yes, you can write 12 as 3*4 or 2*6, but you can continue either way and eventually reach the unambiguous 2*2*3. ↩