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Are there infinitely many pairs of rational numbers $(a,b)$ such that $a^3+1$ is not a square in $\mathbf{Q}$, $b^3+2$ is not a square in $\mathbf{Q}$ and $b^3+2 = x^2(a^3+1)$ for some $x$ in $\mathbf{Q}$?

This question can be phrased also as follows:

Are there infinitely many rational numbers $(a,b)$ such that the extensions $\mathbf{Q}(\sqrt{a^3+1})$ and $\mathbf{Q}(\sqrt{b^3+2})$ are quadratic and equal?

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    What motivated this question? It seems like this should have some relation to elliptic curves.2012-11-26

1 Answers 1

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Yes. Consider the two elliptic curves: $7r^2 = a^3+1$ and $7s^2 = b^3+2.$

One can check (using SAGE or MAGMA) that these two curves have infinitely many rational points (in fact they both have rank 1).

For rational points $(a,r)$ and $(b,s)$ on these curves we have that $a$ and $b$ satisfy the requirements of your question.

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    I like this answer$a$lot. It prompts the following question: Are there infinitely many square-free integers $n$ such that $nr^2=a^3+1$ has positive rank and $ns^2=b^3+2$ has positive rank?2012-11-27