Quadratic Legendre-symbol correlation

For every odd prime

p

,

\sum_{x \in F_{p}} (\frac{x ( x + 1 )}{p}) = - 1.

Equivalently, count the

p - 1

solutions of

t^{2} = y^{2} - 1

by factoring

(y - t) (y + t) = 1

.

Table of contents
- Number of consecutive nonzero quadratic residues modulo an odd prime Quadratic Legendre-symbol correlation

Number of consecutive nonzero quadratic residues modulo an odd prime

The number of

z \in {1, \dots, p - 2}

for which both

z

and

z + 1

are quadratic residues modulo

p

is

\frac{p - 4 - ( - 1 ) ^{(p - 1) /2}}{4} .

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