What are the integer solutions of $x^2+1=2y^3$? I believe that (1,1) is the only solution.
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1Please show what you tried, as it stays it is an isolated problem without any context. See also https://math.meta.stackexchange.com/questions/9959/how-to-ask-a-good-question - Providing context is a must, it improves the quality of the question, makes searches simpler, and answers will not repeat things you already know. Also, giving more words exposes the own view, level, and strategy to attack. Answers can then come pointed and provide in this manner the help that makes the progress in the thinking process. – dan_fulea Nov 25 '24 at 14:12
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If $(x_0,y_0)$ is a solution, then $(-x_0,y_0)$ is also a solution. – dan_fulea Nov 25 '24 at 14:17
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It is similar to Euler's equation, see this post. Actually, here, is a duplicate, for $N=2$. – Dietrich Burde Nov 25 '24 at 14:22