Proof of some conjectures involving quadratic residues

2019/07/30 by Fedor Petrov, Zhi‐Wei Sun, Petrov, Fedor +1
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.1907.12981

Abstract

We confirm several conjectures of Sun involving quadratic residues modulo odd primes. For any prime p≡ 1\pmod 4 and integer a\not≡0\pmod p, we prove that \beginalign*&(-1)^|\1≤ k\ak²\_p\| &+|\(j,k): 1≤ j\frac p2\| ≡&|\1≤ k\δT_k\_p\|, where T_m denotes the triangular number m(m+1)/2.

Related

• Products of quadratic residues and related identities
• Some determinants involving quadratic residues modulo primes
• Quadratic residues that are not primitive roots
• Applications of Lerch's theorem and permutations concerning quadratic residues
• On some determinants arising from quadratic residues