The technical details of RSA works on the idea that it is easy to generate modulus by multiplying two sufficiently large prime numbers together, but factorizing number back into original extremely difficult. Suppose \(N=p^r q^s\) are modulus, where \(p\) and \(q\) product unknown unbalance primes for \(2 \leq s<r\). paper proves using an approximation \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{...

Forward We start with the set of natural numbers, N = {1, 2, 3, . . .} equipped with the familiar addition and multiplication and assume that it satisfies the induction axiom. It allows us to establish division with a residue and the Euclid’s algorithm that computes the greatest commond divisor of two natural numbers. It also leads to a proof of the fundamental theorem of arithmetic: Every natu...

we investigate the classical h.~zassenhaus conjecture‎ ‎for integral group rings of alternating groups $a_9$ and $a_{10}$ of degree‎ ‎$9$ and $10$‎, ‎respectively‎. ‎as a consequence of our‎ ‎previous results we confirm the prime graph‎ ‎conjecture for integral group rings of $a_n$ for all $n leq 10$‎.

BACKGROUND Sodium/hydrogen exchanger-1 (NHE-1) contributes to maintaining intracellular pH (pHi). We assessed the effect of glucose, insulin, leptin and adrenaline on NHE-1 activity in human monocytes in vitro. These cells play a role in atherogenesis and disturbances in the hormones evaluated are associated with obesity and diabetes. METHODS AND RESULTS Monocytes were isolated from 16 health...

The purpose of this paper is to introduce some new classes of rings and modules that are closely related to the classes of almost Dedekind domains and almost Dedekind modules. We introduce the concepts of $\phi$-almost Dedekind rings and $\Phi$-almost Dedekind modules and study some properties of this classes. In this paper we get some equivalent conditions for $\phi$-almost Dedekind rings and ...

in this paper we study the concept of ph-biatness ofa banach algebra a, where ph is a continuous homomorphism on a.we prove that if ph is a continuous epimorphism on a and a hasa bounded approximate identity and a is ph- biat, then a is ph-amenable. in the case where ph is an isomorphism on a we showthat the ph- amenability of a implies its ph-biatness.

Let $R$ be an arbitrary ring and $T$ be a submodule of an $R$-module $M$. A submodule $N$ of $M$ is called $T$-small in $M$ provided for each submodule $X$ of $M$, $Tsubseteq X+N$ implies that $Tsubseteq X$. We study this mentioned notion which is a generalization of the small submodules and we obtain some related results.

Let $M_R$ be a module with $S=End(M_R)$. We call a submodule $K$ of $M_R$ annihilator-small if $K+T=M$, $T$ a submodule of $M_R$, implies that $ell_S(T)=0$, where $ell_S$ indicates the left annihilator of $T$ over $S$. The sum $A_R(M)$ of all such submodules of $M_R$ contains the Jacobson radical $Rad(M)$ and the left singular submodule $Z_S(M)$. If $M_R$ is cyclic, then $A_R(M)$ is the unique ...