let $r$ be a commutative ring with identity and $m$ be an$r$-module. let $fspec(m)$ denotes the collection of all prime fuzzysubmodules of $m$. in this regards some basic properties of zariskitopology on $fspec(m)$ are investigated. in particular, we provesome equivalent conditions for irreducible subsets of thistopological space and it is shown under certain conditions$fspec(m)$ is a $t_0-$spa...

In this work, by employing the Krasnosel'skii fixed point theorem, we study the existence of positive solutions of a three-point boundary value problem for the following fourth-order differential equation begin{eqnarray*} left { begin{array}{ll} u^{(4)}(t) -f(t,u(t),u^{prime prime }(t))=0 hspace{1cm} 0 leq t leq 1, & u(0) = u(1)=0, hspace{1cm} alpha u^{prime prime }(0) - beta u^{prime prime pri...

Is it possible that the lengths of two sides of a primitive Heron triangle have a c ommon factor? The triangle with sides 9, 65, 70 and area 252 shows that this is possible. But notice that the common factor 5 is a prime of the form 4+1. Surprisingly, however, such a common factor cannot be a prime of the form 4 ; 1. Heron's name is familiar to anyone who has used the formula = p s(s ; a)(s ; b...

Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian...

In this paper, new algorithms to solve certain special instances of the Discrete Logarithm Problem (DLP) is presented. These instances are generally considered hard in literature. If a cryptosystem is based on a prime p such that p − 1 is either 2q with q a prime; or 2ρ where ρ = γ1γ2 . . . γkq with γs being small prime factors and q a large prime factor, and the exponent is chosen in the middl...

The first seventeen even perfect numbers are therefore obtained by substituting these values of ra in the expression 2n_1(2n —1). The first twelve of the Mersenne primes have been known since 1914; the twelfth, 2127 —1, was indeed found by Lucas as early as 1876, and for the next seventy-five years was the largest known prime. More details on the history of the Mersenne numbers may be found in ...