Find all the primes that satisfy $p \mid 2^p - 1$

Question

I just want somebody to verify that my work is correct.

Find all prime numbers $p$ such that $p \mid 2^p - 1$.

My claim is that there no such prime that satisfies this.

If $pk = 2^p - 1$, we must have $2^p - 1 \equiv 0 \pmod{p}$. By using Fermat's little theorem we can deduce that $2^p \equiv 2 \pmod{p}$, and so we must also have $2^p - 1 \equiv 2 - 1 \equiv 1 \equiv 0 \pmod{p}$. But $p$ is prime, so $p$ can't divide $1$, and consequently there is no no prime $p$ that will satisfy the above.

since we have $2^p - 1$ $\equiv$ 0 (mod p) but $2^p$ $\equiv$ 2 so we have $2^p$ - 1 $\equiv$ 1 $\equiv$ 0 (mod p) — Dude, Apr 20 '15 at 02:08
It's what I get for posting from my phone. The proof looks good. — Michael Burr, Apr 20 '15 at 02:14
I would suggest putting the $\equiv 0$ closer to the $2^p-1$ to indicate that it comes from the $p\mid 2^p-1$ and not from the other calculations. — Michael Burr, Apr 20 '15 at 02:19

score 1 · Accepted Answer · edited Apr 13 '17 at 12:19

Your argument is correct. $\,n\nmid 2^n-1\,$ for $\,n>1\,$ more generally, by the following

Theorem $\,\ \ \ m,n>1,\,\ m\mid 2^{\large n}-1\,\Rightarrow\, \ell(m) > \ell(n),\ \ $ $\ell(x) =\,$ least prime factor of $\,x $

Proof $\quad\ \, {\rm mod}\,\ p = \ell(m)\!:\,\ 2^{\large n}\equiv 1\equiv\, 2^{\large\color{}{ p-1}}\,$ $\Rightarrow$ $\,\ \ell(m) = p > {\rm ord}\,2\color{#80f}{ \ge \ell(n)}$

Remark $\,\ $ The $ $ key dea $ $ is: $\ 2^n\equiv 1,\,\ \color{#0a0}{2\not\equiv 1}\,$ implies the order of $\,2\,$ is $\,\color{#80f}{\ge\ {\rm least\ prime}\,\ q\mid n},\ $ since the order must divide $\,n,\,$ and is $\color{}{\neq \color{#c00}1}\,$ (else $\,\color{#0a0}{2^{\color{#c00}{\large 1}}\equiv 1}),\,$ i.e. the least nontrivial divisor of $\,n\,$ is its least prime factor (by uniqueness of prime factorizations).

Find all the primes that satisfy $p \mid 2^p - 1$

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