Consecutive Pairs of Quadratic Residues/Examples/61

Examples of Consecutive Pairs of Quadratic Residues

There are $15$ consecutive pairs of quadratic residues modulo $61$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac {61} 4}$.

Proof

From Quadratic Residues modulo $61$:

$\set {1, 3, 4, 5, 9, 12, 13, 14, 15, 16, 19, 20, 22, 25, 27, 34, 36, 39, 41, 42, 45, 46, 47, 48, 49, 53, 56, 57, 58, 60}$ are the quadratic residues modulo $61$

The set of pairs of consecutive quadratic residues modulo $61$ is therefore:

$\set {\set {3, 4}, \set {4, 5}, \set {12, 13}, \set {13, 14}, \set {14, 15}, \set {15, 16}, \set {19, 20}, \set {41, 42}, \set {45, 46}, \set {46, 47}, \set {47, 48}, \set {48, 49}, \set {56, 57}, \set {57, 58}, \set {60, 1} }$

The result follows.

$\blacksquare$

Sources

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