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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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 11H / iii / Solution

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