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The problem statement is: Find all positive integers $n$ such that $\frac{2^{n-1}+1}{n}$ is integer.

Source: LTE Amir Houssein Pavardi P.25

My approach:

It is clear that $n=1$ is a solution, so supose that $n>1$.

$2^{n-1}\equiv -1$ $mod$ $n$;

$2^{2(n-1)}\equiv 1$ $mod$ $n$;

Observe that $n$ is odd hence $n-1$ is even. But if $p=ord_n(2)$ then $p|2(n-1)$ but not $(n-1)$ so $p|2$ how ever it is a contradiction because $n-1$ is even. So the unique solution is $n=1$

Is this proof correct? Because I'm not sure if order properties does apply to composite modulo. And if anyone has a solution involving LTE trick please post a partial proof Thanks in advance!

Weijie Chen
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  • Why should $p \mid 2(n-1)$ and $p \nmid n - 1$ imply $p \mid 2$? – Mr. Chip Aug 07 '16 at 12:08
  • @JoshuaCiappara By orders properties (at least for prime modulo, that's why I asked) because if $a^d\equiv 1$ $mod$ $n$ if and only if $ord_n(a)\mid d$. It can be seen with the property that $gcd(a^b-1,a^d-1)=a^{gcd(b,d)}-1$. – Weijie Chen Aug 07 '16 at 12:19
  • Are you assuming $\mathrm{ord}_n(2)$ is prime, perhaps? $6\mid 2\cdot 9$ but it is not true that $6\mid 9$. Still, it is also not true that $6\mid 2$. – Thomas Andrews Aug 07 '16 at 12:20
  • @ThomasAndrews No, I'm not, but from the properties of the orders I think that they should by either primer or 1. If you read my last comment you will understand, because if $p\nmid d$ then $2^d\not\equiv 1$ $mod$ $n$ – Weijie Chen Aug 07 '16 at 12:22
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    Why should the order be prime? The order of 2 mod 5 is 4, which isn't prime. – Dylan Aug 07 '16 at 14:24
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    Related : http://math.stackexchange.com/questions/1397224/show-that-n-does-not-divide-2n-1-where-n-is-an-integer-greater-than-1 and http://math.stackexchange.com/questions/770256/can-an-odd-number-n-divide-2n-1 – lab bhattacharjee Aug 07 '16 at 15:08
  • @Dylan Ups your right! I'm idiot I'll try again. – Weijie Chen Aug 07 '16 at 16:09

1 Answers1

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It is clear that $n$ must be odd, and hence every prime factor of $n$ is odd. Suppose that $n \neq 1$. Then $n$ has some odd prime factor.

Of all of the prime factors of $n$, let $p$ be the one for which the maximum power of $2$ which divides $p - 1$ is smallest.

Then we can write $p = 2^s \cdot t + 1$ for some $s > 0$ and odd natural number $t$. By the way we chose $p$, we have that for any prime factor $q$ of $n$ that $q \equiv 1 \pmod{2^s}$, and hence that $n \equiv 1 \pmod{2^s}$.

Thus we can write $n = 2^s \cdot \alpha + 1$ for some natural number $\alpha$. Let $g$ be a primitive root modulo $p$. Then there is some natural number $\beta$ such that $g^\beta \equiv 2 \pmod p$. Now we know that $$p \,\mid\, n \,\mid\, 2^{n-1} + 1$$ and so $$ g^{\beta(n-1)} \equiv 2^{n-1} \equiv -1 \equiv g^{(p-1)/2} \pmod p. $$

It follows that $$ 2^s \alpha \beta \equiv \beta(n - 1) \equiv \frac{p-1}{2} \equiv 2^{s-1} t \pmod{(p-1)} $$ and so $$ 2^s \alpha \beta \equiv 2^{s - 1} t \pmod{2^s t} $$ which implies that $$ 0 \equiv 2^{s - 1} t \pmod{2^s} $$ which is a contradiction since $t$ is odd.

Dylan
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  • Why $q\equiv 1$ $mod$ $2^s$? it is clear that every prime is $q\equiv 1$ $mod$ $4$. – Weijie Chen Aug 09 '16 at 12:43
  • For every prime $q$ which divides $n$, there is a largest power of $2$ which divides $q - 1$. We choose $p$ to be the prime for which that power of $2$ is the smallest. So $p - 1$ is divisible by $2^s$, and for every other prime $q$, the largest power of $2$ which divides $q - 1$ is at least $2^s$. So the claim that every $q$ satisfies $q \equiv 1 \pmod{2^s}$ follows from how we choose $p$. Alternatively, you could define $s$ to the largest natural number such that $q \equiv 1 \pmod{2^s}$ for every prime $q$ which divides $n$. – Dylan Aug 09 '16 at 12:50
  • Ok, and last thing can you explain me why? $$ 2^s \alpha \beta \equiv \beta(n - 1) \equiv \frac{p-1}{2} \equiv 2^{s-1} t \pmod{(p-1)} $$ – Weijie Chen Aug 09 '16 at 12:55
  • $g$ is an element of order $p - 1$. (Such an element exists for every prime $p$.) This means that since $g^{\beta (n - 1)} \equiv g^{(p-1)/2}$ that $\beta (n - 1) - (p - 1)/2$ must be divisible by $p - 1$ since $g^{\beta(n - 1) - (p-1)/2} \equiv 1 \pmod p$. But we also have that $\beta (n - 1) = 2^s \alpha \beta$ and that $(p - 1)/2 = 2^{s - 1} t$. – Dylan Aug 09 '16 at 12:58
  • Finally the last two staments why does the first implies the second one? – Weijie Chen Aug 09 '16 at 13:04
  • We have that $2^s \alpha \beta \equiv 2^{s - 1} t \pmod {2^s t}$. Since $2^s ,\mid, 2^s t$, this implies that $2^s \alpha \beta \equiv 2^{s - 1} t \pmod {2^s}$. But $2^s \alpha \beta \equiv 0 \pmod {2^s}$, so this implies that $2^{s-1} t \equiv 0 \pmod {2^s}$. – Dylan Aug 09 '16 at 13:06