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While reading this question regarding OEIS sequence A006517 listing integers $n$ with $n\mid 2^n+2$, I tried to understand the proof that if $n\mid 2^n+2$ and $n>1$, then $n$ is even. This proof in the COMMENTS section of A006517 is:

If an odd term $n>1$ exists then $n = m2^k + 1$ for some $k\ge1$ and odd $m$. Then $n$ divides $2^{m2^k} + 1$ and so does every prime factor $p$ of $n$, implying that $2^{k+1}$ divides the multiplicative order of $2$ modulo $p$ and thus $p-1$. Therefore $n = m2^k + 1$ is the product of prime factors of the form $t2^{k+1} + 1$, implying that $n-1$ is divisible by $2^{k+1}$, a contradiction.

What I don't understand is the part in bold character.

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The order mentioned in the bold part is obviously a divisor of $m\cdot 2^{k+1}$. Suppose, the order is even a divisor of $m\cdot 2^k$. Then, we have $$2^{m2^k}\equiv 1\mod p$$ which is a contradiction to $$2^{m2^k}\equiv -1\mod p$$ since $p$ must be odd. Hence, the order divides $m\cdot 2^{k+1}$ , but not $m\cdot 2^k$. This is only possible, if the order is a multiple of $2^{k+1}$

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  • $\begingroup$ Thank you, I think I understand now. The order is a divisor of $m \cdot 2^{k+1}$ because $(2^{m2^k}+1)(2^{m2^k}-1)=2^{m2^{k+1}}-1$ right? $\endgroup$ – mbjoe Mar 13 at 22:36
  • $\begingroup$ No, because of $$2^{m\cdot 2^{k+1}}=(2^{m\cdot 2^k})^2\equiv (-1)^2=1\mod p$$ $\endgroup$ – Peter Mar 13 at 22:38
  • $\begingroup$ But I just notice that your argument is also valid. $\endgroup$ – Peter Mar 13 at 22:41

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