Economists use circular reasoning all the time but I’ll digress from economics in this post on something I came across recently involving Fermat’s Last Theorem.
There’s a proof of the irrationality of 21/n , where n is an integer for n > 2 (proof doesn’t work for n = 2) which goes something like this:
Suppose 21/n = p/q, where p and q are integers and n > 2 . Then:
pn = qn + qn
violating Fermat’s last theorem. So 21/n is irrational by contradiction.
Sounds cool. But not so. It’s circular argument. A comment at mathoverflow by a person named JS Milne points that that Andrew Wiles’ proof of Fermat’s Last Theorem itself uses the irrationality of 21/n .