Euclid’s proof of the infinitude of the primes is a paragon of simplicity: given a finite list of primes, multiply them together and add one. The resulting number, say N , is not divisible by any prime on the list, so any prime factor of N is a new prime. Some special cases of Dirichlet’s theorem admit a simple proof following Euclid’s model, such as the case of 1 mod 4 or 5 mod 6. (We mean by ...

Let D ≡ 7 mod 8 be a positive squarefree integer, and let hD be the ideal class number of ED = Q( √−D). Let d ≡ 1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k ≥ 0 there is a constant M = M(k), independent of the pair (D, d), such that if (−1)k = sign(d), (2k + 1, hD) = 1, and √ D > 12 π d(log |d|+ M(k)), then the central L-value L(k + 1, χ D,d ) > 0. Furthermore, ...

In this paper, we study dynamical properties as shadowing and structural stability for a class of functions on the p -adic metric spaces Z Q where ? 2 is prime integer.

Abstract. We define an Artin prime for an integer g to be a prime such that g is a primitive root modulo that prime. Let g ∈ Z \ {−1} and not be a perfect square. A conjecture of Artin states that the set of Artin primes for g has a positive density. In this paper we study a generalization of this conjecture for the primes produced by a polynomial and explore its connection with the problem of ...

In this paper, a novel architecture for a finite field integer multiplier based on a superset of the triple moduli set {2-1,2,2+1} is presented. The superset is constructed by adding extra moduli of the form 2±1 (for any non-zero integer n) to extend its range. The multiplier uses index transform approach whereby multiplication is carried out by index addition, and is organized into submodules ...

We extend the Siegel-Walfisz theorem to a family of integer sequences that are characterized by constraints on size prime factors. Besides powers, this includes smooth numbers, almost primes and practical numbers.

Prime generating polynomial functions are known that can produce sequences of prime numbers (e.g. Euler polynomials). However, polynomials which produce consecutive prime numbers are much more difficult to obtain. In this paper, we propose approaches for both these problems. The first uses Cartesian Genetic Programming (CGP) to directly evolve integer based prime-prediction mathematical formula...

Let k = p11p a2 2 · · ·p am m be the prime factorization of a positive integer k and let bk(n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let Sk(N ;M) be the number of positive integers n ≤ N for which bk(n) ≡ 0 (mod M). If pi i ≥ √ k, we prove that, for every positive integer j lim N→∞ Sk(N ; p j i ) N = ...