Wikipedia:Reference desk/Archives/Mathematics/2021 June 4

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June 4

An elementary proof is there

How to show that "For the equation $x^{n}-y^{2}=1$, it's impossible to have integral values for "x" & "y" when "n" is any positive integer>1".? — Preceding unsigned comment added by Rajesh Bhowmick (talk • contribs)

I have cleaned up the formatting of your question for you. Apparently the result you mention (a special case of Catalan's conjecture) was proved by Victor-Amédée Lebesgue in 1850. Our article on Catalan's conjecture gives the reference Victor-Amédée Lebesgue (1850), "Sur l'impossibilité, en nombres entiers, de l'équation $x^{m}=y^{2}+1$", Nouvelles annales de mathématiques, 1re série, 9: 178–181. The paper is available from Numdam, here. --JBL (talk) 18:25, 4 June 2021 (UTC)

You need to require that $y\neq 0.$ Otherwise $x=1$ solves the equation for all $n.$  --Lambiam 21:57, 4 June 2021 (UTC)

For even 'n' it is quite obvious that the problem has only a trivial solution. Ruslik_Zero 15:50, 5 June 2021 (UTC)