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

In particular when the difference is 4, there are 8 sets of three-terms AP.

Can anybody give a hint how can I prove this?

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?