The Most Beautiful Equation in the World

And the Geometry of Numbers

This is part two of a series about Leonard Euler (1707–1783). You can read the first part here.

Before we begin our quest towards understanding the beautiful fact called Euler’s Identity, let’s warm up with an amazing piece of history.

Numbers in Love

About 500 b.c. the Greeks regarded some numbers as more important than others. In particular, they knew of two numbers with a remarkable property. The two numbers are 220 and 284.

Before explaining why these numbers are so interesting, we need to know what a proper divisor is. Well, it is very simple. A proper divisor of say, n, is a natural number smaller than n, that divides it. So for example, the proper divisors of 6 are 1, 2, and 3. Now, the reason that the two numbers above are interesting is that the sum of the proper divisors of 220 is 284 and the sum of the proper divisors of 284 is 220. This relationship is called amicability and the numbers are called amicable numbers (amicable meaning friends or lovers). In fact, it used to be a tradition for two lovers to pick up a fruit, write one of those two numbers on one half of the fruit, the other number on the other half of the fruit, divide the fruit into the corresponding two halves, and then consuming a piece each. This would “unite them and their love forever”.

The Greeks regarded this as a very important relationship but they couldn’t find any more of such numbers no matter how hard they tried. It remained that way for about a thousand years until Thābit ibn Qurra found two more pairs in the 9th century. Back in those days, the center of mathematics had moved from Europe and Egypt to the Arabic world where it would remain for almost half a millennium.

Thabit’s discovery along with further progress in e.g. Iran was however not carried to Europe where only one pair (from the Greeks) was known. That was until Fermat found a pair in 1636. The amicable numbers he found was 17,296 and 18,416.

In this period, there was a mathematical civil war going on between two mathematical giants. Namely, Pierre De Fermat and René Descartes. They hated each other and now Fermat had found a pair of amicable numbers, therefore Descartes…