Let R be a prime ring of char R = 2, d a nonzero derivation of R, ρ a nonzero right ideal of R and 0 = b ∈ R such that b[[d2[x, y], [x, y]]n = 0 for all x, y ∈ ρ, n ≥ 1 fixed integer. If [ρ, ρ]ρ = 0 then either bρ = 0 or d(ρ)ρ = 0. Mathematics Subject Classification: 16W25, 16R50, 16N60

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 prim...

New Result EREW PRAM: Compute gcd(x, y) with probability 1 − o(1) in O(n log log n/ log n) time using n6+ processors. [16] Reduction Our inputs are integers x, y with x ≥ y > 0. • Choose a prime bound B > 0, and assume p | x or p | y implies p > B. • Choose a at random, 1 ≤ a ≤ y − 1. • Compute r := ax mod y. • Remove all prime divisors ≤ B from r producing s. Thus P (r/s) ≤ B. We use (x, y) → ...

Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted A|B if there’s a K such that KA = B. We say that P is prime if for A > 0, A|P only for A = 1 and A = P . Modulu For every two numbers A and B there is unique K and R such that 0 ≤ R ≤ B − 1 and A = KB + R. In this case we say that R = A (mod P ). Cle...

in this paper, we define prime (semiprime) hyperideals and prime(semiprime) fuzzy hyperideals of semihypergroups. we characterize semihypergroupsin terms of their prime (semiprime) hyperideals and prime (semiprime)fuzzyh yperideals.

Let q denote a large prime number, a be any integer. Let N(q, a) denote the smallest positive integer n for which φ(n) ≡ a (mod q). The number N(q, a) exists. Indeed, if a+1 ≡ 0 (mod q) then one can take n = q. Otherwise, one can take n to be a prime from the arithmetical progression a + 1 (mod q). The problem of upper bound estimates for N(q, a) has been a subject of study of the work of Fried...

It is well known that no nonconstant polynomial with integer coefficients can take on only prime values. We isolate the property of the integers that accounts for this, and give several examples of integral domains for which there are polynomials that only take on unit or prime values. Throughout this note, we let R denote an arbitrary UFD (unique factorization domain) that is not a field. Of c...

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...

For a prime polynomial f(T ) ∈ Z[T ], a classical conjecture predicts how often f has prime values. For a finite field κ and a prime polynomial f(T ) ∈ κ[u][T ], the natural analogue of this conjecture (a prediction for how often f takes prime values on κ[u]) is not generally true when f(T ) is a polynomial in T p (p the characteristic of κ). The explanation rests on a new global obstruction wh...