In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted (h, q)-extension of Euler polynomials and numbers, by using p-adic q-deformed fermionic integral on Zp. By applying their generating functions, they derived the complete sums of products of the twisted (h, q)-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new q-extension o...

In this paper, we show that for any positive integer k, the set {( φ(Fn+1) φ(Fn) , φ(Fn+2) φ(Fn) , . . . , φ(Fn+k) φ(Fn) )

• An arithmetic function takes positive integers as inputs and produces real or complex numbers as outputs. • If f is an arithmetic function, the divisor sum Df(n) is the sum of the values of f at the positive divisors of n. • τ (n) is the number of positive divisors of n; σ(n) is the sum of the positive divisors of n. • The Möbius function μ(n) is 1 if n = 1 and 0 if n has a repeated prime fac...

An efficient formulation for the robust shape optimization of aerodynamic objects is introduced in this paper. The formulation has three essential features. First, an Euler solver based on a second-order Godunov scheme is used for the flow calculations. Second, a genetic algorithm with binary number encoding is implemented for the optimization procedure. The third ingredient of the procedure is...

In this paper, the construction of Euler systems of cyclotomic units in a general global function fields is explained. As an application, an analogue of Gras' conjecture in a global function field is proved. 1. I n t r o d u c t i o n and n o t a t i o n s Kolyvagin [7] introduced his remarkable Euler systems to prove important new results on ideal class groups of nmnber felds. This method was ...

Let φ(·) denote the Euler function, and let a > 1 be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as an ≡ 1 (mod φ(n)), and we estimate the number of positive integers n ≤ x satisfying these conditions. 2000 Mathematics Subject Classification. 11A07, 11N37.

We discuss the smallest algebraic number field which contains the nth roots of unity and which may be reached from the rational field Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this field over Q is φ(m), where m is the smallest multiple of n divisible by each prime factor of φ(m). The prime factors of m/n are precisely the primes not dividing n but which do d...

A graph G(V,E) with vertex set V is said to have a prime labeling if there exist a bijection f : V (G) → {1, 2, . . . , |V |} such that for each edge xy ∈ E(G), f(x) and f(y) are relatively prime. It becomes an interesting problem to investigate the maximum possible edges that can be constructed in a graph with n vertices having prime labeling property. We interpret this problem in number theor...