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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$\begingroup$ Related: math.stackexchange.com/questions/874226/… $\endgroup$– ArthurDec 14, 2014 at 0:04
$\begingroup$ See e.g. thm 3.4 @ math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf $\endgroup$– Grigory MDec 14, 2014 at 0:08
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