CTK Insights

Theorem

The following proof has been submitted by Richard Ehrenborg, University of North Carolina, Charlotte and published in the American Mathematical Monthly (May 2003, 423) with a remark that the proof was found by William Henry Schultz, at the time an undergraduate at UNC-Charlotte.

The proof has been reproduced in a recent book from MAA (C. Alsina, R. B. Nelsen, Charming Proofs, p. 36.)

Curiously, the proof relies on Fermat's Last Theorem. Not that the theorem was lacking in elementary proofs, but using the FLT made it into an elegant math joke.

Proof

Assuming that $\sqrt[n]{2}$ is rational: $\sqrt[n]{2} = p/q,$ where both $p$ and $q$ are positive integers. Rewrite this identity as $q^n + q^n = p^n.$ Now by a result of Andrew Wiles, we know that there are no such $p$ and $q.$

Related posts:

1. Trigonometric Proof of Pythagorean Theorem
2. Escher's Theorem
3. Helly's Theorem
4. Nested radicals
5. Radical Simplification