Euler's Amicable Numbers - William Dunham

February 26, 2014 -Princeton University Mathematics Department Colloquium
In this talk, I'll first sketch the life and work of Leonhard Euler (1707 -- 1783), one of the great figures from the long and glorious history of mathematics. I then consider a specific problem from number theory: the construction of amicable pairs (recall that two whole numbers are amicable if each is the sum of the proper whole number divisors of the other). The Greeks knew the amicable pair 220 and 284, and two other pairs were found prior to the 18th century when Euler arrived on the scene. In an awesome display of mathematical power, he found 58 new ones. My mission is to show how he did it -- i.e., how he single-handedly increased the world's supply of amicable numbers twenty-fold. His argument is clever yet so easy to follow that I'll generate a "new" amicable pair right before your eyes. This provides another reminder, if another is necessary, of why Euler is such a towering figure in the history of mathematics.