<s>[INST] Find $m$ so that $(m+4\cdot41)(m^2+4^2\cdot41^2)$ is a square number 
Question: Find the minimum positive odd interger $m$ so that $(m+4\cdot41)(m^2+4^2\cdot41^2)$ is a square number.

Any suggestion will be appreciated. Thanks.
 [/INST] HINT:
Since $m$ is an odd positive integer, so we must have
$$(m+4\cdot 41)(m^2+4^2\cdot 41^2)\equiv 1 \pmod 8$$
$$m^3+4^3\cdot 41^3 + m\cdot 4\cdot 41(m+4\cdot 41) \equiv 1 \pmod 8$$
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