The problem follows,

What is the smallest composite number n, such that $n \vert 2^n-2$

Initially, I thought if we frame the question such that $2^n$ is congruent to $2 \mod n$, we can use the phi function. Through Fermat's little theorem we know that $2^{\phi(n)}$ is congruent to $1 \pmod n$. Thus I tried finding numbers such that the $\phi(n)\vert n$. However, the answer to this problem is 341 and it doesn't fall under the restrictions I gave it.

What am I missing?

  • $\begingroup$ $2^{\phi(n)}$ is 1 mod n, not 2 $\endgroup$ – Bogdan Simeonov Apr 29 '17 at 18:28

We know that $341$ is the smallest base-2 Fermat pseudoprime, i.e., the smallest composite number $n$ satisfying $2^{n-1}\equiv 1 \bmod n$, i.e., satisfying $2^n\equiv 2\bmod n$. So the answer is indeed $n=341=11\cdot 31$ and all "restrictions" are satisfied.


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