Complexity of Computing Quadratic Nonresidues

This note provides new methods for constructing quadratic nonresidues in finite fields of characteristic p. It will be shown that there is an effective deterministic polynomial time algorithm for constructing quadratic nonresidues in finite fields.Comment: References and Improvement

Generalized Fibonacci Primitive Roots

This note generalizes the Fibonacci primitive roots to the set of integers. An asymptotic formula for counting the number of integers with such primitive root is introduced here.Comment: Twelve Pages. Keywords: Primitive root; Fibonacci primitive root; Costas array. arXiv admin note: substantial text overlap with arXiv:1504.00843, arXiv:1405.016

Primitive Roots In Short Intervals

Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $\varepsilon>0$ is a small number. In particular, the least primitive root $g(p)= O\left ((\log p)^{1+\varepsilon} \right)$, and the least prime primitive root $g^*(p)= O\left ((\log p)^{1+\varepsilon} \right)$ unconditionally.Comment: Twenty Pages. Keywords: Least primitive root, Least prime primitive root, Primitive root in short interval. arXiv admin note: substantial text overlap with arXiv:1707.06517 and text overlap with arXiv:1609.01147 and arXiv:1910.0230

Rapidly Convergent Series of the Divisors Functions

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results of some special values of the sums of divisors functions.Comment: Seven Pages. Keywords: Divisors Functions; Rapidly Convergent Series; Multiperfect Number

Spectral Methods And Prime Numbers Counting Problems

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof of the more general dePolignac conjecture on the existence of infinitely many primes pairs p and p + 2k, k => 1, is proposed in this note.Comment: Thirty Pages. Keywords: Distribution of Primes, Twin Primes Conjecture, dePolignac Conjecture, Germain Primes, Prime Diophantine Equation

Twin Primes In Quadratic Arithmetic Progressions

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the existence of infinitely many quadratic twin primes $n^{2}+1$ and $n^{2}+3$, $n \geq 1$, are proposed in this note.Comment: twenty Pages. Keywords: Distribution of Primes, Twin Primes Conjecture, Quadratic Prime

On Evaluation of Nonlinear Exponential Sums

This paper provides a technique for evaluating some nonlinear Gaussian sums in closed forms. The evaluation is obtained from the known values of simpler exponential sums.Comment:

Formulas For The Square Root Modulo p

A method of constructing specific polynomial representations f(x) over the finite field F_p of the square roots function modulo a prime p = 2^k*n + 1, n odd, is presented. The formulas for the cases k = 2, 3 and 4 are given.Comment: 4 Page

Generalized Artin Primitive Root Conjecture

An asymptotic formula for the number of integers with the primitive root 2, and a generalized Artin primitive root conjecture for composite integers is presented here.Comment: Twelve Pages. Keywords: Primitive root; Artin primitive root conjectur

An Explicit Formula For The Divisor Function

The details for the construction of an explicit formula for the divisors function d(n) = #{d | n} are formalized in this article. This formula facilitates a unified approach to the investigation of the error terms of the divisor problem and circle problem.Comment: Twenty Eight Pages. Keywords: Divisor Function, Explicit Formula, Divisor Proble