Integer values of $n$ when $n^3+1$ is a perfect square

Asked May 01 '23 at 16:36

Active May 01 '23 at 16:54

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If $n^3+1$ is a perfect square, find all integer values of $n$. I have proved that when n³+1 is even then the only solution for n is $-1$. However, I am not so confident for my solution in the case of $n^3+1$ being odd. I would appreciate answers solving this specific case. The solutions are $-1,0,2$.

Note: I am aware that this question has been asked already however the solution makes use of elliptic curves and I would prefer a quite elementary and simple solution.

edited May 01 '23 at 16:54

asked May 01 '23 at 16:36

Aarav Gupta

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Context for this question? – Mike May 01 '23 at 16:39
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Writing $n^3=(m-1)(m+1)$ we see that both $m\pm 1$ must be perfect cubes (possibly ignoring powers of $2$). – lulu May 01 '23 at 16:43
@Mike I found this on a video online on math Olympiad problems. They did not provide the solution. – Aarav Gupta May 01 '23 at 16:45
@lulu Yes but I want to prove that if n³ is even then the only values for n are 0,2. – Aarav Gupta May 01 '23 at 16:46

Integer values of $n$ when $n^3+1$ is a perfect square

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