Track / Stage 5 / 32 of 400 #632 of 1964

Problem 632

AIME late

Number theory Difficulty 5.1 Find the answer

4 Try to find a set of positive integer solutions for the equation $x^{2}-51 y^{2}=1$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

$\begin{array}{l} \text { Hint: } x^{2}=51 y^{2}+1 \\ =49 y^{2}+14 y+1+2 y^{2}-14 y \\ =\left(7 y+1\right)^{2}+2 y(y-7) \\ \end{array}$

Let $y=7$ . Then $x=50$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.