# The secret of squares revealed

We count in sets of ten. This seems natural to us because we have ten fingers. However the ancient Babylonians used different units, which is why we measure time in units of 60 minutes and clock-faces have 12 hours. We need not use sets of ten, any number would do. Mathematicians call this modular arithmetic. So we count in modulus ten. When perfect squares and modular arithmetic are combined strange and unexpected things happen. A question mathematicians have wanted to answer for hundreds of year is this: when is a number a perfect square when units are counted in a prime number modulus? (A prime is a special number that can only be divided by 1 and itself, e.g. 2, 3, 5, 7, 11 and so on.) It turns out that the relationship between a number and its square when the counting units are a prime is so surprising that mathematicians have been trying to decide what it means for hundreds of years.

* *philosophy mathematics measurement decimal Fermat nature