About 500 BCE 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 had to find another. In 1638 he finds the pair 9,363,584 and 9,437,056. This, I remind you, is without a calculator! It must have been some long rainy days.
It turns out that these two pairs that Fermat and Descartes had found were the same pairs that Thābit had found.
Thus, the status of amicable numbers remained at only three known pairs after 2000 years of bright minds and rainy days…
Then Euler decides to give it a try.
Euler finds 58 more pairs of amicable numbers!
That is wildly insane. Of course, what happened was not brute-force trial and error. Instead, Euler found a method relying on properties of the sum-of-divisor function as well as some genius insights.
Are there infinitely many, you ask? No one knows… This is again, one of the mysteries of mathematics.