We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.

the notions of quasi-prime submodules and developed  zariski topology was introduced by the present authors in cite{ah10}. in this paper we use these notions to define a scheme. for an $r$-module $m$, let $x:={qin qspec(m) mid (q:_r m)inspec(r)}$. it is proved that $(x, mathcal{o}_x)$ is a locally ringed space. we study the morphism of locally ringed spaces induced by $r$-homomorphism $mrightar...

We prove that for integers a > b > c > 0, the greatest prime factor of (ab+1)(ac+1) tends to infinity with a. In particular, this settles a conjecture raised by Györy, Sarkozy and Stewart, predicting the same conclusion for the product (ab + 1)(ac + 1)(bc + 1). In the paper [GSS], Gÿory, Sarkozy and Stewart conjectured that, for positive integers a > b > c, the greatest prime factor of the prod...

In this paper, we find all integers x such that x − 1 has only prime factors smaller than 100. This gives some interesting numerical corollaries. For example, for any positive integer n we can find the largest positive integer x such that all prime factors of each of x, x + 1, . . . , x+ n are less than 100.

We prove that whenever A and B are dense enough subsets of {1, . . . , N}, there exist a ∈ A and b ∈ B such that the greatest prime factor of ab + 1 is at least N1+|A|/(9N).

It is proved that every odd perfect number is divisible by a prime greater than 107.

It is proved here that every odd perfect number has a prime factor greater

Letting P k (n) stand for the k-th largest prime factor of n ≥ 2 and given an irrational number α and a multiplicative function f such that |f (n)| = 1 for all positive integers n, we prove that n≤x f (n) exp{2πiαP k (n)} = o(x) as x → ∞.

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding 107. Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding 108.

It is proved that every odd perfect number is divisible by a prime greater than 107.