Quadratic residues and non-residues in arithmetic progression

On quadratic non-residues

The Distribution of Quadratic Residues and Non-residues

1. If p is a prime other than 2, half of the numbers 1, 2, ..., p-l are quadratic residues (modp) and the other half are quadratic non-residues. Various questions have been proposed concerning the distribution of the quadratic residues and non-residues for large p, but as yet only very incomplete answers to these questions are known. Many of the known results are deductions from the inequality ...

Abelian groups and quadratic residues in weak arithmetic

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S 2 + iWPHP(Σ b 1), and use it to derive Fermat’s little theorem and Euler’s criterion for the Legendre symbol in S 2 + iWPHP(PV ) extended by the pigeonhole principle PHP(PV ). We ...

Quadratic Residues and Their Application

This project explores quadratic residues, a classical number theory topic, using computational techniques. First I conducted computational experiments to investigate the distribution of quadratic residues modulo primes, looking for patterns or evidence against randomness. The experimental data indicates a non-random distribution of quadratic residues. Certain features of such non-random distrib...

Camel sequences and quadratic residues

Given an even cyclic (+1; 1) sequence s = (s0; :::; s2n 1) which consists of n plus ones and n minus ones, let us compute i+ j (mod 2n) for all n2 (+1; 1)-pairs (si; sj) and distribute obtained n2 numbers between 2n "boxes" 1; 2; :::; 2n 1; 2n = 0 (mod 2n). The average cardinality of a box is n2 2n = n2 . Some sequences have quite remarkable box-distributions, "almost average everywhere with tw...