2000 IMO Problems/Problem 5

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Does there exist a positive integer $n$ such that $n$ has exactly 2000 prime divisors and $n$ divides $2^n + 1$?

Solution

Let $N=2^n+1$. We will assume for the sake of contradiction that $n|N$.

$2^n+1 \equiv 0$ (mod $n$) $\Rightarrow 2^n \equiv -1$ (mod $n$). So 2 does not divide $n$, and so $n$ is odd.

Select an arbitrary prime factor of $n$ and call it $p$. Let's represent $n$ in the form $p^am$, where $m$ is not divisible by $p$.

Note that $p$ and $m$ are both odd since $n$ is odd. By repeated applications of Fermat's Little Theorem:

$N = 2^n+1 = 2^{p^am} + 1 = (2^{p^{a-1}m})^p + 1 \equiv 2^{p^{a-1}m} + 1$ (mod $p$)

Continuing in this manner, and inducting on k from 1 to $a$,

$2^{p^{a-k}m}+1 \equiv (2^{p^{a-k-1}m})^p + 1$ (mod $p$) $\equiv 2^{p^{a-k-1}m} + 1$ (mod $p$)

So we have $N \equiv 2^m+1$ (mod $p$)

Since $p$ is relatively prime to $m$, $N \equiv 1+1$ (mod $p$) $\equiv 2$ (mod $p$)

Since $p$ is odd, $N$ is not divisible by $p$. Hence $N$ is not divisible by $n$. So we have a contradiction, and our original assumption was false, and therefore $N$ is still not divisible by $n$.

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