Sums Involving Quadratic Residues Modulo a Prime of the Form p = 4n + 1

摘要

The sum of quadratic residues for a prime p = 4n + 1 is p(p -1)/2. An alternative formula is obtained based on some observed patterns. Inspired by the terms that appear on this formula, we introduce the concept of total residue and jump. A very detailed analysis of these jumps is done which allows us to find the sum of residues of k~2 + k + 1 — n (modulo p) from 0 up to n, 2n and 4n. Among many other formulas, one for the sum of quadratic residues (modulo p) from 1 to n is also obtained.