A number b divides a if the remainder is zero. We denote it by b | a. b a denotes that b does not divide a. If b divides a then a is a multiple of b. Now we can define the greatest common divisor (GCD). The GCD of two numbers a and b is defined as the biggest number which divides both a as well as b. It is also denoted by gcd(a, b). One of the important case is when gcd(a, b) = 1, i.e., there i...

Let p, q be distinct primes satisfying gcd(p−1, q−1) = d and letDi, i = 0, 1, · · · , d−1, be Whiteman’s generalized cyclotomic classes with Z∗ pq = ∪ d−1 i=0Di. In this paper, we give the values of Gauss periods based on the generalized cyclotomic sets D∗ 0 = ∑ d 2 −1 i=0 D2i and D ∗ 1 = ∑ d 2 −1 i=0 D2i+1. As an application, we determine a lower bound on the 2-adic complexity of modified Jaco...

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. x 2 Z n with minimum max 1in jxij satisfying P n i=1 xiai = gcd(a1; : : : ; an). We present a polynomial-time algorithm which approximates a minimum gcd set for a1; : : : ; an within a factor 1+ln n and prove that this algorithm is best possible in the se...

a two dimensional generalized thermoelastic problem of a thick circular plate of finite thickness and infinite extent subjected to continuous axisymmetric heat supply and an internal heat generation is studied within the context of generalized thermoelasticity. unified system of equations for classical coupled thermoelasticity, lord-shulman and green-lindsay theory is considered. an exact solut...

This note is a response to one of the problems posed by Kwa´sniewski in [1, 2], see also [3] i.e. GCD-morphic Problem III. We show that any GCD-morphic sequence F is at the point product of primary GCD-morphic sequences and any GCD-morphic sequence is encoded by natural number valued sequence satisfying condition (C1). The problem of general importance-for example in number theory was formulate...

Received Jul 17 th , 2012 Accepted Aug 26 th , 2012 Recently, many fully-homomorphic encryption schemes have been constructed. However, the issue of the security of these fully homomorphic encryptions has not been carefully studied. By using lattice reduction algorithm, we firstly present an attack on the fully homomorphic encryption based on approximate GCD over the integers. Our result shows ...

We give a brief survey on some classical and recent results concerning the generalized Lebesgue-Ramanujan-Nagell equation. Moreover, we solve completely the equation x + 1117 = y in nonnegative integer unknowns with n ≥ 3 and gcd(x, y) = 1. 1 Generalized Ramanujan-Nagell equations Mixed polynomial-exponential equations are of classical and recent interest. One of the most famous equation of thi...