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I have a tutorial question(not homework), that asks to prove that there exists no $x,y \in \mathbb{Z}$, solution for:

$$x^5 - 3y^5 = 2008$$

I originally thought I would solve it by taking all cases $\mod 10$, but realised that would be a massive workload, and also realised it fell through instantly as $x = 1 \mod 1, y = 1 \mod 1$, gave $x^5 - 3y^5 = 8 \mod 10$ instantly. I only thought that method might work because we were doing mod proofs directly prior to the question.

I have absolutely no idea how to solve this problem without using computation, any tips or methods?

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If $\pmod {10}$ doesn't work you could try other moduli. Maybe the $3$ in the equation makes you think of $3^n$. I would try $\pmod 9$ The fact that $2008 \equiv 0 \pmod 8$ seems interesting as well. No guarantees. –  Ross Millikan Mar 29 at 3:41
    
Why do you think the workload would be so massive? You don't actually need to do every pair - for a given prime $p$ just compute a table of all the fifth powers, mod $p$, and then a table of three times all the fifth powers; checking to see whether the shift of any given number shows up on the other is basically an $O(1)$ operation, and so you should be able to check modulo any given prime $p$ in $O(p)$ steps pretty easily. This isn't great, but it's not bad at all. –  Steven Stadnicki Mar 29 at 3:42
    
@RossMillikan Unfortunately neither of those suggestions works; every odd number is a fifth power mod $8$, and $2008$ is $1$ mod $9$. –  Mike Miller Mar 29 at 3:49
    
@Mike: Playing a bit more, also every number is a fifth power mod $9$. It seemed like a good shot to me. To me, it would be the next try. Looks like Dan Brumleve had a better thought than mine. –  Ross Millikan Mar 29 at 4:04
    

1 Answer 1

up vote 8 down vote accepted

Hint: Try mod $11$ instead, noting that both $x^5$ and $y^5$ are in $\{-1,0,1\}$. The intuition behind choosing $11$ is Fermat's little theorem (all tenth powers mod $11$ are either $0$ or $1$).

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