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I don't know the exact source of this question, but here it goes:

Find all ordered pairs $(x,y)$ such that $x,y \in \Bbb{N}$ and satisfy the relation: $$y^3=x^2+2$$

Of course $(x,y)=(5,3)$ is a solution but proving that infinitely many exist or none other exist is the part I'm stuck at.

I tried to work an infinite descent with congruent modulo but due to difference in degree one coefficient always vanishes. I am not that good at olympiad number theory, so do help me out here.

Shikhar_Mohan
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  • I do not see that $(3,5)$ is a solution. Did you mean $(5,3)$? – A.Γ. Jan 08 '17 at 12:25
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    You meant $(5,3)$, presumably. Are you familiar with properties of algebraic integers? – lulu Jan 08 '17 at 12:26
  • No, not really. Could you point me in the direction of a resource? – Shikhar_Mohan Jan 08 '17 at 12:29
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    This is a fairly straightforward question about the ring of integers $\mathbb Z[\sqrt {-2}]$. this ring arises because in it your expression factors as $y^3=(x-\sqrt {-2})(x+\sqrt {-2})$. You'll want to know that this ring is a unique factorization domain, which is a fairly standard argument but not trivial if you've never seen it. I'll dig up some good references. – lulu Jan 08 '17 at 12:31
  • After having found one solution you can write $y^3-3^3=x^2-5^2$. – A.Γ. Jan 08 '17 at 12:35
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    here is a proof that this ring is a ufd. this contains a solution to your question. Note, in the last reference, that the OP is asking a similar but different question. However one of the solvers, @AndreNicolas, gives a very clear solution to your question. Again, the argument is standard if you are used to algebraic number fields, but will take some time to read if you are not. – lulu Jan 08 '17 at 12:36
  • @A.G. I've tried that. I used congruent modulo to eliminate the integers and ended up with a bi-variable equation, which did nothing. – Shikhar_Mohan Jan 08 '17 at 12:43
  • @lulu All the NT I've done is olympiad level so I don't have the foggiest about analytical number theory. Do you know of any resources that could help me use this tool? Thanks for your help. – Shikhar_Mohan Jan 08 '17 at 12:45
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    This is algebraic number theory, no analytics involved. this link contains a number of solid recommendations for introductory texts. – lulu Jan 08 '17 at 12:48
  • @lulu Thanks a lot! – Shikhar_Mohan Jan 08 '17 at 12:50

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