Consecutive quadratic residues

On Consecutive Quadratic Non-residues: a Conjecture of Issai Schur

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest possible number of consecutive quadratic non-residues modulo p is always less than p. (One can find a brief discussion of this problem in R. K. Guy’s book [5]). Schur also pointed out that the greatest number of consecutive q...

On quadratic non-residues

Pairs of Consecutive Power Residues

Introduction. Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l , . . . , r + m — 1, each of which is a &th power residue of p. Let...

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...

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...