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If $(a,p)=1$ and $p$ is an odd prime, prove the Legendre symbol sum

$$\sum _{n=1}^ p \left(\frac{an+b}{p}\right)=0.$$ Where $b$ is any integer.

I know the fact that $\sum_{a=1}^p \left( \frac{a}{p} \right)=0$. But I don't know how to treat with $b$.

Martin Sleziak
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Martin
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  • Welcome to math.SE: I have tried to improve the readability of your question by introducing MathJaX. It is possible that I unintentionally changed the meaning of your question. Please proofread the question to ensure this has not happened. – gebruiker Nov 15 '15 at 17:46
  • the brackets are required because this is not a fraction but the legendre symbol. – Brennan.Tobias Feb 01 '16 at 15:33
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    See also http://math.stackexchange.com/questions/26212/legendre-symbol-showing-that-sum-m-0p-1-left-fracambp-right-0 and http://math.stackexchange.com/questions/1666704/a-problem-with-the-legendre-jacobi-symbols-sum-n-1p-left-fracanbp – Martin Sleziak Feb 22 '16 at 14:11

1 Answers1

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Hint: Since $\gcd(a,p) = 1$, it follows that as $n$ ranges from $1$ to $p$, $an$ takes on each residue class from $0$ to $p-1$ modulo $p$. Can you take it from there?

rogerl
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