On amicable numbers

Abstract

Translated from the Latin original, "De numeris amicabilibus" (1747). E100 in the Enestroem index. Euler starts by saying that with the success of mathematical analysis, number theory has been neglected. He argues that number theory is still a worthwhile subject. First, because there are very elegant and difficult questions in it. Second, because some of the greatest figures in mathematics so far had spent time working on problems in number theory. In particular, Descartes spent time looking for (and finding) a pair of amicable numbers that previously in Europe had been unknown. Euler also mentions the work of Frans van Schooten. Then Euler explains the form of the amicable numbers that were then known, which were $2^n xy$ and $2^n z$, where $x,y,z$ are prime, and the two conditions i) $z=xy+x+y$ and ii) $2^n(x+y+2)=xy+x+y+1$ are satisfied. Euler says that to find more amicable numbers different forms should be tried. He gives 30 pairs of amicable numbers but does not explain how he found them. This is discussed in Ed Sandifer's November 2005 "How Euler did it". One of the pairs (pair XIII) Euler lists is incorrect.