Competition problem — Number theory

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Number theory Difficulty 4.6 AIME Find the answer

For which integers $n \in\{1,2, \ldots, 15\}$ is $n^{n}+1$ a prime number?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

$n=1$ works. If $n$ has an odd prime factor, you can factor, and this is simulated also by $n=8$ : $a^{2 k+1}+1=(a+1)\left(\sum_{i=0}^{2 k}(-a)^{i}\right)$ with both parts larger than one when $a>1$ and $k>0$ . So it remains to check 2 and 4, which work. Thus the answers are $1,2,4$ .

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