2n is irrational for n3 (proof using Fermat’s last theorem)

Theorem 1.

If n3, then 2n is irrational.

The below proof can be seen as an example of a pathological proof. It gives no information to “why” the result holds, or how non-trivial the result is. Yet, assuming Wiles’ proof does not use the above theoremMathworldPlanetmath anywhere, it proves the statement. Otherwise, the below proof would be an example of a circular argument.


Suppose 2n=a/b for some positive integers a,b. It follows that 2=an/bn, or

bn+bn = an. (1)

We can now apply a recent result of Andrew Wiles [1], which states that there are no non-zero integers a, b satisfying equation (1). Thus 2n is irrational. ∎

The above proof is given in [2], where it is attributed to W.H. Schultz.


  • 1 A. Wiles, Modular elliptic curves and Fermat’s last theorem, Annals of Mathematics, Volume 141, No. 3 May, 1995, 443-551.
  • 2 W.H. Schultz, An observation, American Mathematical Monthly, Vol. 110, Nr. 5, May 2003. (submitted by R. Ehrenborg).
Title 2n is irrational for n3 (proof using Fermat’s last theorem)
Canonical name sqrtn2IsIrrationalForNge3proofUsingFermatsLastTheorem
Date of creation 2013-03-22 13:38:32
Last modified on 2013-03-22 13:38:32
Owner matte (1858)
Last modified by matte (1858)
Numerical id 13
Author matte (1858)
Entry type Proof
Classification msc 11J72