A simple proof that is irrational:

Assume that it’s rational. Then = *p*/*q* and *p*^{2} = 2*q*^{2}. But in the latter equation, the left side must have an even number of prime factors and the right an odd number. That’s a contradiction, so our assumption must be wrong.

In the paper below, Manchester Polytechnic mathematician T.J. Randall credits this “marvellous” proof to Philip J. Davis and Reuben Hersh in their 1982 book *The Mathematical Experience*, but I can’t find it there.

(T.J. Randall, “67.45 Revisited,” *Mathematical Gazette* 67:442 [December 1983], 302-303.)

12/20/2023 UPDATE: Reader Hans Havermann finds the proof mentioned in Stuart Hollingdale’s *Makers of Mathematics* (1989), after the more familiar proof based on parity of *p* and *q*. Hollingdale writes that the alternative proof “can be traced back to the Classical period.”