IMC 2024 · Problem 10

We say that a square-free positive integer $n$ is almost prime if

$$n \mid x^{d_1} + x^{d_2} + \ldots + x^{d_k} - kx$$

for all integers $x$, where $1 = d_1 < d_2 < \ldots < d_k = n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m} + 1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$.

(An integer $n$ is called square-free if it is not divisible by $d^2$ for any integer $d > 1$.)