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Prove that if $(a_n)$ and $(c_n)$ ($n \in \mathbb{N}$) are strictly increasing positive integer sequences with $a_n^2 \mid c_n^2+1$, then the sequence defined by $b_n = c_n+a_n^2c_n-a_n^3$ is strictly increasing.

This result holds if $c_n-a_n$ is increasing since $c_n+a_n^2c_n-a_n^3 = c_n+a_n^2(c_n-a_n)$. What if it isn't?

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  • $\begingroup$ still false with $a_1 = 100, a_2 = 200, c_1 = 200, c_2 = 201$ $\endgroup$ – mercio Sep 15 '16 at 20:37
  • $\begingroup$ Just an observation, $a_n^2 | c_n^2 + 1$ is equivalent to the negative Pell equation $c_n^2 - k a_n^2 = -1$. This drastically limits the range of possible values $a_n, c_n$ and may help with the proof (though I don't see it offhand). $\endgroup$ – dxiv Sep 15 '16 at 23:44
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The claim is false.

If $a_1=1, a_2=3, c_1=1, c_2=2$, we have $b_1=1$ and $b_2=-7$.

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  • $\begingroup$ I changed the question. $\endgroup$ – user19405892 Sep 15 '16 at 23:36

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