One might conjecture that there is an interesting fact concerning each of the positive integers. Here is a ‘proof by induction’ that such is the case. Certainly, 1, which is a factor of each positive integer, qualifies, as do 2, the smallest prime; 3, the smallest odd prime; 4, Bieberbach’s number; etc. Suppose the set S of positive integers concerning each of which there is no interesting fact is not vacuous, and let k be the smallest member of S. But this is a most interesting fact concerning k! Hence S has no smallest member and therefore is vacuous. Is the proof valid?

— Edwin F. Beckenbach, “Interesting Integers,” American Mathematical Monthly, April 1945