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Number Theory

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Moderate

Statement Verification: Jacobi Symbol

NUMTH-SL6FSV

Let $a$ be an integer and $n\geq 3$ be an odd integer with $(a,n)=1$.

Which of the following statements about Jacobi symbol are correct? Select ALL that apply.

A

$\left(\frac an\right)=1$ implies that $x^2 \equiv a \ \mathrm{mod} \ n$ has a solution.

B

$\left(\frac an\right)=-1$ implies that $x^2 \equiv a \ \mathrm{mod} \ n$ does not have a solution.

C

If $x^2 \equiv a \ \mathrm{mod} \ n$ has a solution, then $\left(\frac an \right) =1$.

D

If $x^2 \equiv a \ \mathrm{mod} \ n$ does not have a solution, then $\left(\frac an \right)=-1$.