<p>Let <img alt="$p$" height="11" src="https://latex.artofproblemsolving.com/3/6/f/36f73fc1312ee0349b3f3a0f3bd9eb5504339011.png" width="10" /> be the least prime number for which there exists a positive integer <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> such that <img alt="$n^{4}+1$" height="18" src="https://latex.artofproblemsolving.com/d/a/2/da274809f7f098374a3ebc1d4f65ddacc7edb02b.png" width="52" /> is divisible by <img alt="$p^{2}$" height="20" src="https://latex.artofproblemsolving.com/5/a/b/5ab6025259a9dd6a0fea6fc48a34ef3b4ed8028e.png" width="18" />. Find the least positive integer <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> such that <img alt="$m^{4}+1$" height="18" src="https://latex.artofproblemsolving.com/f/7/5/f7522bcf18c4287afb04a5db78cfc6110997b6f3.png" width="57" /> is divisible by <img alt="$p^{2}$" height="20" src="https://latex.artofproblemsolving.com/5/a/b/5ab6025259a9dd6a0fea6fc48a34ef3b4ed8028e.png" width="18" />.</p>

Question No: 427

Let $p$ be the least prime number for which there exists a positive integer $n$ such that $n^{4}+1$ is divisible by $p^{2}$ . Find the least positive integer $m$ such that $m^{4}+1$ is divisible by $p^{2}$ .

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Tags: Number Theory

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