1. We say that g(n) is additive if g(mn)=g(m)+g(n) holds for every coprime pairs m, n of positive integers . If, moreover, g(p') =g(p)' for every prime power pa , then g(n) is called strongly additive . By p, pi , p2 , . . ., q, qr , q2 , . . . we denote prime numbers, c, Cl, c2i . . . are suitable positive constants . F(n) and x(n) denote the largest and the smallest prime factor of n . The sy...

Recently Yehuda Rav has given the concept of Semi prime ideals in a general lattice by generalizing the notion of 0-distributive lattices. In this paper we study several properties of these ideals in a general nearlattice and include some of their characterizations. We give some results regarding maximal filters and include a number of Separation properties in a general nearlattice with respect...

We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisor P(τ(n)) and the number of distinct prime divisors ω(τ(n)) of τ(n) for various sequences of n. In particular, we show that P(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and P(τ(p)τ(p2)τ(p3)) > (1+o(1)) log log p log log log p loglog log log p for every prime p with τ(p) 6= 0.

we prove the generalized hyers--ulam stability  of n--th order linear differential equation of the form $y^{(n)}+p_{1}(x)y^{(n-1)}+ cdots+p_{n-1}(x)y^{prime}+p_{n}(x)y=f(x)$, with condition that there exists a non--zero solution of corresponding homogeneous equation. our main results extend and improve the corresponding results obtained by many authors.

In the late eighteenth century both Euler and Legendre noticed that n +n+41 is prime for n = 0, 1, 2 . . . 39, and remarked that there are few polynomials with such small degree and coefficients that give such a long string of consecutive prime values. Rabinowitsch, at the 1912 International Congress of Mathematicians [18], showed that n + n + A is prime for n = 0, 1, 2, . . . A − 2 if and only...

The search for first occurrences of prime gaps and maximal prime gaps is extended to 1015. New maximal prime gaps of 806 and 906 are found, and sixty-two previously unpublished first occurrences are found for gaps varying from 676 to 906.

For nonnegative real numbers α, β, γ , A, B, and C such that B C > 0 and α β γ > 0, the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . . . has a unique positive equilibrium. A proof is given here for the following statements: 1 For every choice of positive parameters α, β, γ , A, B, and C, all solutions to the difference equation xn 1 α βxn γxn−1 / A Bxn Cxn−1 , n 0, 1, 2, . ...

For any integer n ≥ 1, let j(n) denote the Jacobsthal function, and ω(n) the number of distinct prime divisors of n. In 1962 Jacobsthal conjectured that for any integer r ≥ 1, the maximal value of j(n) when n varies over N with ω(n) = r is attained when n is the product of the first r primes. We show that this is true for r ≤ 23 and fails at r = 24, thus disproving Jacobsthal’s conjecture.

Consider multisets A in the group G = (Z/nZ) such that no non-empty subset has sum zero. It is known for long that the maximal cardinality of such a set is 2n − 2, and there is a conjecture of Gao and Geroldinger describing the structure of such sets of maximal cardinality (called “property B”). Recently, Gao, Geroldinger and Grynkiewicz showed that it is enough to prove the conjecture for n pr...

Abstract In this paper, we call a finite group G G an N L M NLM -group ( C NCM -group, respectively) if every non-normal cyclic subgroup of prime order or 4 (prime power order, in is contained maximal . Using the property -groups and -groups, give new necessary suffic...