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Im trying to prove that an odd primitive root mod an odd prime is also a primitive root for two times that modulus.

Im just not sure exactly how to go about this. I've tried expressing $a^k\equiv1 modP$ as $a^k=1+Pk$ to try to reach an equation that makes sense for 2P. I just feel like Im not going about this the right way.

Ryan H.
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1 Answers1

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Let $a$ be a primitive root modulo $p$ for a odd prime $p$. A integer $a$ is a primitive root modulo $m$ if $$a^{\varphi(m)} \equiv 1 \pmod m$$ and $\varphi(m)$, the number of integers less than $m$ and coprime with $m$, is the smallest positive integer for what this equation holds. As a result of Euler's Theorem, we have:

$$a^{\varphi(p)} \equiv 1 \pmod p$$

Because we know that $a$ is a primitive root modulo $p$, we know that $\varphi(p)$ is the smallest positive integer for which the equation holds. By hypothesis, $a$ is odd, so we also know that

$$a^{\varphi(p)} \equiv 1 \pmod 2$$

Thus, by the Chinese Remainder Theorem, we must have that:

$$a^{\varphi(p)} \equiv a^{\varphi(2p)} \equiv 1 \pmod{2p}$$

Note that first equality follows from multiplicative property of the $\varphi$ function: $\varphi(2p) = \varphi(p)$ for odd primes $p$. Now, suppose that, for a positive integer $n$, we have:

$$a^n \equiv 1 \pmod{2p}$$

By the Chinese Remainder Theorem, that would imply:

$$a^n \equiv 1 \pmod{p}$$

We know that the least such integer is $\varphi(p)$, so $n \geq \varphi(p)$. Therefore, $n \geq \varphi(2p)$, and that concludes the proof.

Henrique Augusto Souza
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  • This isn't correct. $a$ itself doesn't have to be coprime with $\phi(p)=p-1$. There are lots of primes for which $a=2$ is a primitive root, for example. – Greg Martin Jun 01 '16 at 03:39
  • @GregMartin you are correct, I forgot about that. It must be coprime with $\varphi(\varphi(p))$ right? Because of the group isomorphism. I'll correct it now. – Henrique Augusto Souza Jun 01 '16 at 03:50
  • @GregMartin I believe that the algebraic proof is correct now. I have also missed the odd root hypothesis, which made me even the more wrong. – Henrique Augusto Souza Jun 01 '16 at 04:01
  • The first part is still not correct (again, think of $a=2$). I recommend deleting it entirely. The second proof is correct, though the wording could be improved. Proving that $a^{\phi(2p)}\equiv1\pmod{2p}$ uses the Chinese remainder theorem and Fermat's little theorem modulo $p$; proving that $a^k\not\equiv1\pmod{2p}$ for any $1\le k\le \phi(2p)$ follows from considering the congruence modulo $p$ and using the fact that $a$ is a primitive root. – Greg Martin Jun 01 '16 at 05:53
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    @GregMartin now I see. The second condition involves the image of the primitive root on the corresponding additive group (the discreet logarithm). On a more general notion, all I needed to do was show that $a$ was the generator of the multiplicative group, and that there was a natural isomorphism between $\mathbb{Z}/2p\mathbb{Z}^\times$ and $\mathbb{Z}/p\mathbb{Z}^\times \times \mathbb{Z}/2\mathbb{Z}^\times$. Between the arithmetic proof and the algebraic proof using that discreet logarithm, I now think that the arithmetic one is more accessible. Thanks for your suggestions! – Henrique Augusto Souza Jun 01 '16 at 06:14
  • Thanks for the help! This makes sense to me now! – Ryan H. Jun 01 '16 at 21:36