Find-the-natural-roots-of-the-equation-x-2-51y-2-1-

Question Number 148418 by mathdanisur last updated on 27/Jul/21

$${Find}\:{the}\:{natural}\:{roots}\:{of}\:{the}\:{equation} \\ $$$${x}^{\mathrm{2}} \:-\:\mathrm{51}{y}^{\mathrm{2}} \:=\:\mathrm{1} \\ $$

Answered by mr W last updated on 27/Jul/21

x^2 −1=51y^2 (x−1)(x+1)=51y^2 x−1=y^2 =7^2 =49 x+1=51 ⇒x=50, y=7

$${x}^{\mathrm{2}} −\mathrm{1}=\mathrm{51}{y}^{\mathrm{2}} \\ $$$$\left({x}−\mathrm{1}\right)\left({x}+\mathrm{1}\right)=\mathrm{51}{y}^{\mathrm{2}} \\ $$$${x}−\mathrm{1}={y}^{\mathrm{2}} =\mathrm{7}^{\mathrm{2}} =\mathrm{49} \\ $$$${x}+\mathrm{1}=\mathrm{51} \\ $$$$\Rightarrow{x}=\mathrm{50},\:{y}=\mathrm{7} \\ $$

Commented by mathdanisur last updated on 27/Jul/21