This is a question French mathematician Pierre de Fermat posed to the English mathematicians of his time. "Prove that x=5 and y=3 are the only positive integer values for which $x^2 +2=y^3$." I have been trying to come up with an answer for days using modular arithmetic to no avail. Any such method would therefore be appreciated as well as other insights.
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Related: http://math.stackexchange.com/questions/874226/proof-that-26-is-the-one-and-only-number-between-square-and-cube – Arthur Dec 14 '14 at 00:04
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See e.g. thm 3.4 @ http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf – Grigory M Dec 14 '14 at 00:08