Maths Olympiad Prep

Track / Stage 5 / 63 of 400 #663 of 1964

Problem 663

AIME late
Number theory Difficulty 5.1 Find the answer

4 Try to find a set of positive integer solutions for the equation x2−51y2=1x^{2}-51 y^{2}=1

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

Next problem →

Official solution

 Hint: x2=51y2+1=49y2+14y+1+2y2−14y=(7y+1)2+2y(y−7)\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=7y=7. Then x=50x=50

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.