Paul Erdős and the rise of statistical thinking in elementary number theory

Abstract

It might be argued that elementary number theory began with Pythagoras who noted two-and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means d │ n and 1 ≤ d < n), then s(220) = 284 and s(284) = 220. When faced with remarkable examples such as this it is natural to wonder hw special they are. Through the centuries mathematicians tried to find other examples of amicable pairs, and they did indeed succeed. But is there a formula? Are there infinitely many? In the first millennium of the common era, Thâbit ibn Qurra came close with a formula for a subfamily of amicable pairs, but it is far from clear that his formula gives infinitely many examples and probably it does not. A special case of an amicable pair m,n is when m = n. That is, s(n) = n. These numbers are called perfect, and Euclid came up with a formula for some of them (and perhaps all of them) that probably inspired that of Thâbit for amicable pairs. Euler showed that Euclid’s formula covers all even perfect numbers, but we still don’t know if Euclid's formula gives infinitely many examples and our knowledge about odd perfects, even whether any exist, remains rudimentary.