  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.
