## Proof that 2^(1/n) is irrational

October 31, 2012 by Shubhendu Trivedi

After an interesting conversation with somebody recently I was looking around and aggregating simple mathematical facts that have somewhat crazy proofs. Like for all such questions, I found a great MathOverflow thread and decided to share this gem from there:

**Fact:** is irrational for any integer .

**Proof:** Suppose it is not. Then , then , or contradicting Fermat’s Last Theorem.

Although a commenter there mentions that the argument is essentially circular (which I find fascinating), but other than that it made me laugh, what I find interesting about this answer and an accompanying comment by Greg Kuperberg is that it made me realize that **Fermat’s Last Theorem is not strong enough to imply the irrationality of **.

### Like this:

Like Loading...

*Related*

on November 14, 2012 at 1:05 am |Rod CarvalhoWhy not link directly to the answer? There are 56 answers in that thread!

on November 14, 2012 at 1:09 am |Shubhendu TrivediI thought I did. A missed copy paste I guess.Thanks for pointing that out. Fixed.