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

= 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 $$y\ne 0.$$ 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)