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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\$.