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a) Let $p$ be an odd prime. Prove that:

$$\text{ord}_p(a)=2 \iff a\equiv -1 \mod p$$

My attempt:

Assume that $\text{ord}_p(a)=2$, then

$a^2\equiv 1 \mod p$

$p\mid a^2-1$

$p\mid(a-1)(a+1)$

$p\mid a-1$ or $p\mid a+1$

If $p\mid a-1$ then $a\equiv 1 \mod p$

Which contradicts that $\text{ord}_p(a)=2$

Therefore, we must have $a\equiv -1 \mod p$.

Now, assume that $a\equiv -1 \mod p$

Then $a^2 \equiv 1 \mod p$. Now, I think we must show that $2$, is the least positive integer satisfying the last congruence. This is equivalent to showing, that $a\not \equiv 1 \mod p$. Since $a\equiv -1 \mod p$, then $a\not \equiv 1 \mod p$. Is that true, please?

b) Suppose that $\text{ord}_n (a)=n-1$, prove that n is a prime number.

My attempt:

$\text{ord}_n (a)=n-1 \implies a^{n-1} \equiv 1 \mod n$

Then and by the converse of the Fermat’s little theorem, we have that $n$ is a prime number. [notice that $(a,n)=1$].

Is that true, please?

Thank you.

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    $\begingroup$ Yes, it could be possible only with $p=2$, and you suppose $p$ is odd. $\endgroup$ – Bernard Mar 30 at 22:54
  • $\begingroup$ @Bernard Thank you so much. $\endgroup$ – Dima Mar 30 at 22:59
  • $\begingroup$ You're welcome! Always glad to help! $\endgroup$ – Bernard Mar 30 at 23:00
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    $\begingroup$ B.t.w., non-prime numbers which satisfy that $a^{n-1}\equiv 1$ for all $a$ coprime to $n$ are called *Carmichael numbers. The smallest Carmichael number is $561=3\cdot 11\cdot 17$. $\endgroup$ – Bernard Mar 30 at 23:10
  • $\begingroup$ " Now, I think we must show that 2, is the least positive integer satisfying the last congruence." uh.... he only positive integer smaller is $1$.... Sometimes one of the directions in an if and only if proof is self evident. This is one of those times. $\endgroup$ – fleablood Mar 31 at 1:10
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Hint For $b)$ your approoach is not working since the converse to FLT is not true.

Try instead the following: $\text{ord}_n (a)=n-1$ implies that $a, a^2,... , a^{n-1}$ are distinct elements modulo n, in the set $\{1, 2, .., n-1\} \pmod{n}$.

Deduce that $1, 2,.., n-1$ are all invertible modulo n

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