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I also wonder about an other interesting residue relation

Let P be a prime,

let [tex] a^{2^n}[/tex] be called a cyclic quadratic residue if there is integer m dependent on [tex]a[/tex] such that [tex] a^{2^{n + mp}} = a^{2^n}[/tex] for all integers [tex]p \mod P[/tex]

It seems that the sum of all such cylic residues is either 0 or 1 mod P

For instance for P = 17 the only cyclic residue is 1 but for P = 37

there are the cyclic sequences

33 16 34 9 7 12 33 ...

10 26 10 ...

1 ...

and the sum of all these numbers, not including repetitions is 4*37.

Let P be a prime,

let [tex] a^{2^n}[/tex] be called a cyclic quadratic residue if there is integer m dependent on [tex]a[/tex] such that [tex] a^{2^{n + mp}} = a^{2^n}[/tex] for all integers [tex]p \mod P[/tex]

It seems that the sum of all such cylic residues is either 0 or 1 mod P

For instance for P = 17 the only cyclic residue is 1 but for P = 37

there are the cyclic sequences

33 16 34 9 7 12 33 ...

10 26 10 ...

1 ...

and the sum of all these numbers, not including repetitions is 4*37.

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