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I am looking for a elementary (and possibly short) prove that

$$y^2=x^3+23$$ has no solutions $(x,y)\in \mathbb Z^2$.

Reducing the equation modulo $p$ didn't help.

Thanks in advance:).

Edit: I want to see a solution without using the theory of elliptic curves. This question appeared in our last exam of "introduction to number theory". We only know basic facts like the legendre symbol, hensels lemma, Chinese Remainder Theorem etc..

Marc
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  • Hello :) it is not elementary enough for me. This question appeared in our test of the course introduction to number theory. No knowledge about elliptic curves was required. – Marc Oct 02 '16 at 08:28
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    for the answer see this here http://math.stackexchange.com/questions/245299/integer-solutions-for-x2-y3-23 – Dr. Sonnhard Graubner Oct 02 '16 at 08:29
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    @Marc: For me, the answer doesn't involve elliptic curves at all. I don't know where you've seen elliptic curves in the proof given in the link… – Watson Oct 02 '16 at 08:35
  • @Watson Mordell curves are special type of elliptic curves, so this has very much to do with elliptic curves. IIRC there are on-line sources listing integral points of Mordell curves. Judging from what I've seen in some cases elementary methods suffice to prove that a list of solutions is comprehensive. Some cases it is not easy. I'm no expert, so I won't say anything definite. – Jyrki Lahtonen Oct 02 '16 at 08:46
  • @JyrkiLahtonen: I agree that this can be related to elliptic curves. But the proof of this particular question given in the link only involves elementary methods, I think. – Watson Oct 02 '16 at 08:48
  • Ok @Watson. Understood. – Jyrki Lahtonen Oct 02 '16 at 08:51

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