Question

Is 2^n + 3^n (where n is an integer) ever the square of a rational number?

Answer 1

2^n + 3^n=x^2

Answer 2

So yes or no?

Answer 3

If you agree with my equation than yes :P

Answer 4

Timo, it is a bit more complicated than that. The theorem as posed is:
\[\exists n\in\mathbb{N} \text{ such that }\sqrt{2^n+3^n}\in\mathbb{Q}.\]As for the answer, I don't know.

Answer 5

it is highly unlikely...but im not sure how to mathematically prove it either way

Answer 6

woah that looks confusing:/ It was a question asked by my friend but i don't get any of this really.

Answer 7

i could run a program...but theoretically there are an infinite possibilities for n
so you couldn't say never...maybe proof by induction

Answer 8

I thought induction at first too, by contradiction, but I think that's a blind alley. @jazy, it's a great question, and is fun to think about. I don't have the tools yet to answer it well, other than to write a program like dumbcow suggested. I'm sure there are ways of thinking about the question that I don't have access to yet that would make it answerable.

Answer 9

I looked it up on google and came accross this:
http://www.qbyte.org/puzzles/p155s.html
I guess thats where she got it from....always trying to trick me!

Answer 10

Yep, I was just looking it up on google as well. Turns out the proof uses modular arithmetic, which I would not have thought of. Cool stuff.

Answer 11

Yup, I guess it's pretty cool to think about...Once I get there:)