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Let p be a prime number of form 4k+1. I guess that there are c(d) number of 3-terms arithmetic progressions (AP) in the set of quadratic residues modulo p, where c(d) is an integer constant depending on the d difference of the progression, i.e. the number of occurrences is independent of p.

In particular when the difference is d = 4, there are c = 8 sets of three-terms AP when p > 17.

Can anybody give a hint how can I prove this?

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    $\begingroup$ It is confusing that $k$ denotes two different things in your question, namely $p=4k+1$ and $k$ is the difference of your AP. It is also confusing that you do not use TeX here. $\endgroup$
    – GH from MO
    Commented Dec 14, 2014 at 21:49

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The statement is false. For example, for the case of $d=1$, it is known that the largest number of consecutive quadratic residues modulo $p$ tends to infinity as $p\to\infty$. See here and, for more general patterns, see Seva's answer to this MO question. Further related MO entries are here and here.

Added. Seva's answer (linked above) shows that the statement is false for every $d$.

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  • $\begingroup$ Fair enough. (I wrote: I guess) Can you say anything about d = 4 ? $\endgroup$
    – Methus
    Commented Dec 14, 2014 at 22:07
  • $\begingroup$ @Methus: See my added section. $\endgroup$
    – GH from MO
    Commented Dec 14, 2014 at 22:20

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