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Number Theory

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Moderate

Properties of Primitive Roots

NUMTH-OJRESV

Let $p$ be an odd prime number. We say that an integer $a$ is a primitive root modulo $p$ if

$$ \{a^k : k=0, 1, \cdots, p-1\}\equiv \{1,2,\cdots p-1\} \ \ (\mathrm{ mod } \ p).$$

Suppose that $a$ is a primitive root modulo $p$. Which of the following statements are true?

Check ALL that apply.

A

$a^{p-1}\equiv 1\ \ (\mathrm{ mod } \ p)$.

B

$a^{\frac{p-1}2}\equiv 1 \ \ (\mathrm{ mod } \ p)$.

C

$a^{p^2+p}\equiv a \ \ (\mathrm{ mod } \ p)$.

D

$a+1$ is a primitive root modulo $p$.

E

$a^{p^{p^p}}\equiv a \ \ (\mathrm{ mod } \ p)$.