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Number Theory

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Properties of Primitive Roots: Quadratic Nonresidue

NUMTH-$VX41S

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.

Note: We say that:

$$\{a_1,\cdots , a_n\}\equiv \{b_1,\cdots, b_n\} \ \ (\mathrm{ mod }\ p)$$

...if the two sets:

$$\{[a_1], \cdots , [a_n]\}, \ \ \{[b_1], \cdots , [b_n]\}$$

...of congruence classes modulo $p$ are identical.

A

$\{a^{jk} : k=0, 1, \cdots, p-1\}\equiv \{1,2,\cdots p-1\} \ \ (\mathrm{ mod } \ p)$ for any integer $j>1$

B

$\{a^{jk} : k=0, 1, \cdots, p-1\}\equiv \{1,2,\cdots p-1\} \ \ (\mathrm{ mod } \ p)$ for some integer $j>1$.

C

The number of primitive roots modulo $p$ is always less than the number of quadratic nonresidues modulo $p$.

D

The number of primitive roots modulo $p$ is always greater than the number of quadratic nonresidues modulo $p$.

E

The number of primitive roots modulo $p$ is sometimes equal to the number of quadratic nonresidues modulo $p$.