Abstract:
We derive, for all prime moduli p except those in a very thin set, an upper bound for the least prime primitive root (mod p) of order of magnitude a constant power of log p. The improvement over previous results, where the upper bound was log p to an exponent tending to infinity with p, lies in the use of the linear sieve (a particular version called the shifted sieve) rather than Brun's sieve. The same methods allow us to rederive a conditional result of Shoup on the least prime primitive root (mod p) for all prime moduli p, assuming the generalized Riemann hypothesis. We also extend both results to composite moduli q, where the analogue of a primitive root is an element of maximal multiplicative order (mod q).

Abstract:
A sieve is constructed for ordinary twin primes of the form 6m+/-1 that are characterized by their twin rank m. It has no parity problem. Non-rank numbers are identified and counted using odd primes p>=5. Twin- and non-ranks make up the set of positive integers. Regularities of non-ranks allow gathering information on them to obtain a Legendre-type formula for the number of twin-ranks near primorial arguments.

Abstract:
Using the properties of the table sieve, we can determine whether all given number, positive integer G, is a prime and whether it is possible to factor it out.

Abstract:
Let G(xt,x) denote the number of Gaussian integers with norm not exceeding x2t whose Gaussian prime factors have norm not exceeding x2. Previous estimates have required restrictions on the parameter t with respect to x. The purpose of this note is to present asymptotic estimates for G(xt,x) for all ranges of the parameter t with respect to x.

Abstract:
A classic question in analytic number theory is to find asymptotics for $\sigma_{k}(x)$ and $\pi_{k}(x)$, the number of integers $n\leq x$ with exactly $k$ prime factors, where $\pi_{k}(x)$ has the added constraint that all the factors are distinct. This problem was originally resolved by Landau in 1900, and much work was subsequently done where $k$ is allowed to vary. In this paper we look at a similar question about integers with a specific prime factorization. Given $\boldsymbol{\alpha}\in\mathbb{N}^{k}$, $\boldsymbol{\alpha}=(\alpha_{1},\alpha_{2},...,\alpha_{k})$ let $\sigma_{\boldsymbol{\alpha}}(x)$ denote the number of integers of the form $n=p_{1}^{\alpha_{1}}... p_{k}^{\alpha_{k}}$ where the $p_{i}$ are not necessarily distinct, and let $\pi_{\boldsymbol{\alpha}}(x)$ denote the same counting function with the added condition that the factors are distinct. Our main result is asymptotics for both of these functions.

Abstract:
We give a more comrehensive treatment of Chen's double sieve and improve related constants in Goldbach's conjecture and the twin prime problem.

This paper will illustrate two versions of an algorithm for finding prime number up to N, which give the first version complexity

(1)

where c1,c2 are constants, and N is the input dimension, and gives a better result for the second version. The method is based on an equation that expresses the behavior of not prime numbers. With this equation it is possible to construct a fast iteration to verify if the not prime number is generated by a prime and with which parameters. The second method differs because it does not pass other times over a number that has been previously evaluated as not prime. This is possible for a recurrence of not prime number that is (mod 3) = 0. The complexity in this case is better than the first. The comparison is made most with Mathematics than by computer calculation as the number N should be very big to appreciate the difference between the two versions. Anyway the second version results better. The algorithms have been

Abstract:
We present a new sieve that allows us to find the prime numbers by using only regular patterns and, more importantly, avoiding any duplication of elements between them.

Abstract:
An s-tuple of positive integers are k-wise relatively prime if any k of them are relatively prime. Exact formula is obtained for the probability that s positive integers are k-wise relatively prime.

Abstract:
We propose a new formulation of the problem of prime factorization of integers. With replica exchange Monte Carlo simulation, the behavior which is seemed to indicate exponential computational hardness is observed. But this formulation is expected to give a new insight into the computational complexity of this problem from a statistical mechanical point of view.