Let g = 0,±1 be a fixed integer. Given two sequences of complex numbers (φm) ∞ m=1 and (ψn) ∞ n=1 and two sufficiently large integers M and N , we estimate the exponential sums ∑ p≤M gcd(ag,p)=1 ∑ 1≤n≤N φpψnep (ag ) , a ∈ Z, where the outer summation is taken over all primes p ≤ M with gcd(ag, p) = 1.

Consider the multivariate polynomial problem over the integers; that is, Gcd(A,B) where A,B ∈ Z[x1, x2, . . . xn]. We can solve this problem by solving the related Gcd problem in Zp[x1, x2, . . . xn] for several primes p, and then reconstructing the solution in the integers using Chinese Remaindering. The question we address in this paper is how fast can we solve the problem Gcd(A,B) in Zp[x1, ...

Downregulation of the fibronectin (FN) gene in a rat 3Y1 derivative cell line, XhoC, transformed by the adenovirus E1A and E1B genes seems to be caused by the induction of a negative regulator, G10BP, which binds to three G-rich sequences in the promoter (T. Nakamura, T. Nakajima, S. Tsunoda, S. Nakada, K. Oda, H. Tsurui, and A. Wada, J. Virol. 66:6436-6450, 1992). These are the G10 stretch and...

We prove that for m < n, the n × m rectangular toroidal chessboard admits gcd(m,n) nonattacking queens except in the case m = 3, n = 6. The classical n-queens problem is to place n queens on the n × n chessboard such that no pair is attacking each other. Solutions for this problem exist for all for n = 2, 3 [1]. The queens problem on a rectangular board is of little interest; on the n ×m board ...

A3 Start with a finite sequence a1, a2, . . . , an of positive integers. If possible, choose two indices j < k such that aj does not divide ak, and replace aj and ak by gcd(aj , ak) and lcm(aj , ak), respectively. Prove that if this process is repeated, it must eventually stop and the final sequence does not depend on the choices made. (Note: gcd means greatest common divisor and lcm means leas...

We show that every [n, k, d]3 code with diversity (Φ0, Φ1), 3 ≤ k ≤ 5, gcd(d, 3) = 1, is (2, 1)-extendable except for the case (Φ0, Φ1) = (40, 36) for k = 5, and that an [n, 5, d]3 code with diversity (40, 36), gcd(d, 3) = 1, is (2, 1)-extendable if Ad ≤ 50. Geometric conditions for the (2, 1)-extendability of not necessarily extendable [n, k, d]3 codes for k = 5, 6 are also given.

In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), highest common factor (hcf), or greatest commonmeasure (gcm), of two or more integers (when at least one of them is not zero), is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.[1][2] This notion can be extended to polynomials, see ...

Let Fq be a finite field with q = p , where p is an odd prime. In this paper, we study the repeated-root self-dual negacyclic codes over Fq. The enumeration of such codes is investigated. We obtain all the self-dual negacyclic codes of length 2p over Fq, a ≥ 1. The construction of self-dual negacyclic codes of length 2bp over Fq is also provided, where gcd(2, b) = gcd(b, p) = 1 and a ≥ 1.