‎let $p$ be a prime with $pgeq 7$ and $q=2(p-1)$‎. ‎in this paper‎ ‎we prove the existence of a nontrivial product of‎ ‎filtration $s+4$ in the stable homotopy groups of spheres‎. ‎this nontrivial‎ ‎product is shown to be represented up to a nonzero scalar by‎ ‎the product element $widetilde{gamma}_{s}b_{n-1}g_{0}in‎ ‎{ext}_{mathcal{a}}^{s+4,(p^n+sp^2+sp+s)q+s-3}(mathbb{z}/p,mathbb{z}/p)$‎ ‎in ...

Monier and Rabin proved that an odd composite can pass the Strong Probable Prime Test for at most 1 4 of the possible bases. In this paper, a probable prime test is developed using quadratic polynomials and the Frobenius automor-phism. The test, along with a xed number of trial divisions, ensures that a composite n will pass for less than 1 7710 of the polynomials x 2 ? bx ? c with b 2 +4c n = ...

The purpose of this paper is to justify the claim that a method for generating primes presented at EUROCRYPT'89 generates primes with virtually uniform distribution. Using convincing heuristic arguments, the conditional probability distributions of the size of the largest prime factor p 1 (n) of a number n on the order of N is derived, given that n satisses one of the conditions 2n+1 is prime, ...

Let Ω(n) and ω(n) denote respectively the total number of prime factors and the number of distinct prime factors of the integer n. Euler proved that an odd perfect number N is of the form N = pem2 where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We prove that Ω(N) ≥ (18ω(N) − 31)/7 and Ω(N) ≥ 2ω(N) + 51.

We study prime divisors of various sequences of positive integers A(n) + 1, n = 1,...,N, such that the ratios a(n) = A(n)/A(n − 1) have some number-theoretic or combinatorial meaning. In the case a(n) = n, we obviously have A(n) = n!, for which several new results about prime divisors of n! + 1 have recently been obtained.

For any closed oriented 3-manifold M $M$ with positive simplicial volume and N $N$ , we prove that there exists a finite cover ′ $M^{\prime }$ of admits degree-1 map f : → $f:M^{\prime }\rightarrow N$ is, virtually 1-dominates . This result generalizes previous virtual domination results hyperbolic domain to more general domains.