Consecutive Pairs of Quadratic Residues/Examples/3

Examples of Consecutive Pairs of Quadratic Residues

There are no Consecutive Pairs of Quadratic Residues modulo $3$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac 3 4}$.

Proof

From Quadratic Residues modulo $3$:

$1$ is a quadratic residue

$2$ is not a quadratic residue.

Hence neither is an element of a pair of consecutive quadratic residues.

The result follows.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {3-5}$ The Use of Computers in Number Theory: Exercise $7$