Consecutive Pairs of Quadratic Residues

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Theorem


Proof


Examples

Consecutive Pairs of Quadratic Residues of $3$

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

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


Consecutive Pairs of Quadratic Residues of $5$

There is $1$ consecutive pair of quadratic residues modulo $5$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac 5 4}$.


Consecutive Pairs of Quadratic Residues of $7$

There is $1$ consecutive pair of quadratic residues modulo $7$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac 7 4}$.


Consecutive Pairs of Quadratic Residues of $11$

There are $2$ consecutive pairs of quadratic residues modulo $11$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac {11} 4}$.


Consecutive Pairs of Quadratic Residues of $17$

There are $4$ consecutive pairs of quadratic residues modulo $17$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac {17} 4}$.


Consecutive Pairs of Quadratic Residues of $29$

There are $7$ consecutive pairs of quadratic residues modulo $29$.

This is consistent with the number of such consecutive pairs being $\floor {\dfrac {29} 4}$.


Consecutive Pairs of Quadratic Residues of $61$

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

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