Summation Series of Legendre Symbols

Asked Dec 02 '16 at 18:36

Active Dec 02 '16 at 19:55

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Let $p$ be an odd prime, and put $$s(a, p) = \sum_{n=1}^{p} (\frac{n(n+a)}{p}) $$

Show that:

(i) $\sum_{a=1}^{p}s(a, p) = 0.$

(ii) If $(a, p) = 1$ then $ s(a, p) = s(1, p).$

(iii) Conclude that $s(a, p) = −1$ if $(a, p) = 1.$

I've tried (i) and I think that the proof relies on $\sum_{n=1}^{p} \frac{n}{p}=0 $, however I don't know how to show this result. (ii) and (iii) I am unsure of as well.

edited Dec 02 '16 at 19:55

asked Dec 02 '16 at 18:36

user3.14259

1

(i) Cannot be true: $n>0\Rightarrow s > 0 \Rightarrow \sum s > 0$ – caverac Dec 02 '16 at 18:57
Again (we discussed this, remember?), the sum is miswritten: Legendre symbol $(a/p)$ is not the fraction $a/p$. The Legendre symbol can be $0$, $1$ or $-1$. For your last question, see this duplicate. – Dietrich Burde Dec 02 '16 at 19:54
Thanks for saying, I've edited it now – user3.14259 Dec 02 '16 at 19:56

Summation Series of Legendre Symbols

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