On finite differences, permutations, and Quadratic residues

On differences of quadratic residues

Factoring an integer is equivalent to express the integer as the difference of two squares. We test that for any odd modulus, in the corresponding ring of remainders, any element can be realized as the difference of two quadratic residues, and also that, for a fixed remainder value, the map assigning to each modulus the number of ways to express the remainder as difference of quadratic residues...

On quadratic non-residues

Power Map Permutations and Symmetric Differences in Finite Groups

Let G be a finite group. For all a ∈ Z, such that (a, |G|) = 1, the function ρa : G → G sending g to g defines a permutation of the elements of G. Motivated by a recent generalization of Zolotarev’s proof of classic quadratic reciprocity, due to Duke and Hopkins, we study the signature of the permutation ρa. By introducing the group of conjugacy equivariant maps and the symmetric difference met...

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...