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Wlod AA
Wlod AA
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Let $p$$\ p\ $ be a prime. Prove that if $p\equiv 3\pmod{4}$$\ p\equiv 3\pmod{4}\ $ then the sum$$S=\sum_{i=0}^{p-1}\left(\frac{i^3+6i^2+i}{p}\right)=0$$What

$$ S=\sum_{k=0}^{p-1}\left(\frac{k^3+6k^2+k}{p}\right)=0 $$

What is the value of the sum $S$$\ S\ $ when $p\equiv 1\pmod{4}$?$\ p\equiv 1\pmod{4}\,?\ $ When $p\equiv 3\pmod{4}$$\ p\equiv 3\pmod{4}\ $ all i know is that $(-1|p)=-1$$\ (-1|p)=-1\ $ but i am not sure if that gives me anything.

Let $p$ be a prime. Prove that if $p\equiv 3\pmod{4}$ then the sum$$S=\sum_{i=0}^{p-1}\left(\frac{i^3+6i^2+i}{p}\right)=0$$What is the value of the sum $S$ when $p\equiv 1\pmod{4}$? When $p\equiv 3\pmod{4}$ all i know is that $(-1|p)=-1$ but i am not sure if that gives me anything.

Let $\ p\ $ be a prime. Prove that if $\ p\equiv 3\pmod{4}\ $ then the sum

$$ S=\sum_{k=0}^{p-1}\left(\frac{k^3+6k^2+k}{p}\right)=0 $$

What is the value of the sum $\ S\ $ when $\ p\equiv 1\pmod{4}\,?\ $ When $\ p\equiv 3\pmod{4}\ $ all i know is that $\ (-1|p)=-1\ $ but i am not sure if that gives me anything.

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GH from MO
GH from MO
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user317834
user317834

Sum of Legendre Symbol when $p\equiv 1,3\mod{4}$

Let $p$ be a prime. Prove that if $p\equiv 3\pmod{4}$ then the sum$$S=\sum_{i=0}^{p-1}\left(\frac{i^3+6i^2+i}{p}\right)=0$$What is the value of the sum $S$ when $p\equiv 1\pmod{4}$? When $p\equiv 3\pmod{4}$ all i know is that $(-1|p)=-1$ but i am not sure if that gives me anything.