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Find the smallest positive integer which can be expressed as the sum of four positive squares, not necessarily different, and divides $2^n + 15$ for some positive integer $n$.

If you let $K$ be that integer you have, $K \equiv 1 \pmod 2$. And if $n \ge 4$ then $16$ divides $K + 1$ but I don't know what to next, hints and solutions would be appreciated.

Taken from the 2009 International World Youth Mathematics Intercity Competition (IWYMIC).

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The smallest odd natural numbers that are sums of four positive squares are $7=4+1+1+1$ and $13=4+4+4+1$. It is easy to show that $2^n+15$ is never divisible by $7$. But $2^7+15=143$ is divisible by $13$.

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  • $\begingroup$ Modulo $7$ the residue class of $2$ generates the subgroup of quadratic residues, and $-15$ is not in that subgroup. But $2$ is a primitive root modulo $13$. $\endgroup$
    – Jyrki Lahtonen
    Commented Oct 7, 2019 at 19:46
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    $\begingroup$ I was going to point out that $23 \equiv 7 \pmod 8$ could not be the sum of three squares, therefore must be the sum of four nonzero squares. Also $23 = 8 + 15$ says we could use 23 dividing itself. So, no larger $\endgroup$
    – Will Jagy
    Commented Oct 7, 2019 at 19:46

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