Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Find the answer

10. Find all solutions to $x^{4}+y^{2}=z^{2}$ satisfying $(x, y)=1$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

10. $x^{2}, y, z$ are the reduced solutions of equation (1). Furthermore, using formulas (6), (7), discuss the solutions of $x^{2}=r^{2}-s^{2}$ or $x^{2}=2 r s$ .

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