The concepts of a prime ideal of a distributively generated (d.g.) nearring R, a prime d.g. near-ring and an irreducible R-group are introduced1). The annihilating ideal of an irreducible R-group with an R-generator is a prime ideal. Consequently we define a prime ideal to be primitively prime if it is the annihilating ideal of such an R-group, and a d.g. near-ring to be a primitively prime nea...

The paper proposes justification of using Mersenne primes in the following cryptoschemes: commutative and publickey encryption algorithms and zero-knowledge protocol. The cryptoschemes are based on computational difficulty of finding discrete logarithm in the finite fields GF (2), where s is a sufficiently large prime such that 2s−1 is also a prime, for example s = 1279, s = 2203, and s = 4253.

let $r$ be a commutative ring with identity and $m$ an $r$-module. in this paper, we associate a graph to $m$, say ${gamma}({}_{r}m)$, such that when $m=r$, ${gamma}({}_{r}m)$ coincide with the zero-divisor graph of $r$. many well-known results by d.f. anderson and p.s. livingston have been generalized for ${gamma}({}_{r}m)$. we show that ${gamma}({}_{r}m)$ is connected with ${diam}({gamma}({}_...

In this article we provide new explicit Chebyshev’s bounds for the prime counting function ψ(x). The proof relies on two new arguments: smoothing the prime counting function which allows to generalize the previous approaches, and a new explicit zero density estimate for the zeros of the Riemann zeta function.

Let λ1, . . . , λ4 be non-zero with λ1/λ2 irrational and negative, and let S be the set of values attained by the form λ1x 3 1 + · · · + λ4x4 when x1 has at most 3 prime divisors and the remaining variables are prime. We prove that most real numbers are close to an element of S.

Let K′ be a knot that admits no cosmetic crossing changes and let C be a prime, non-cable knot. Then any knot that is a satellite of C with winding number zero and pattern K′ admits no cosmetic crossing changes. As a consequence we prove the nugatory crossing conjecture for Whitehead doubles of prime, non-cable knots.

n>0 (1− q) be Dedekind’s eta-function. For a prime p, denote by U Atkin’s Up-operator. We say that a function φ with a Fourier expansion φ = ∑ u(n)q is congruent to zero modulo a power of a prime p, φ = ∑ u(n)q ≡ 0 mod p if all its Fourier expansion coefficients are divisible by this power of the prime; u(n) ≡ 0 mod p for all n. In this paper we prove the following congruences. Theorem 1. (i) I...

In the early stage of optical CDMA systems, the spreading codes, such as optical orthogonal codes (OOCs) and prime codes, are only coded in the time domain. However, the small code size becomes the major issue for one-dimensional codes. To deal with this limitation of onedimensional codes, a new family of two-dimensional codes is proposed for optical CDMA systems. The new codes are modified fro...

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if charac...