Abstract We consider the reduction of an elliptic curve defined over rational numbers modulo primes in a given arithmetic progression and investigate how often subgroup points this reduced is cyclic.

Dirichlet’s theorem on primes in arithmetic sequences states that in any arithmetic progression m,m + k, m + 2k, m + 3k, . . ., there are infinitely many primes, provided that (m, k) = 1. Euler first conjectured a result of this form, claiming that every arithmetic progression beginning with 1 contained an infinitude of primes. The theorem as stated was conjectured by Gauss, and proved by Diric...

For positive integers s and k1,k2, . . . ,ks, the van der Waerden number w(k1,k2, . . . ,ks;s) is the minimum integer n such that for every s-coloring of set {1,2, . . . ,n}, with colors 1,2, . . . ,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this l...

Let G = (V,E) be a graph of order n and size e. An (a, d)-vertexantimagic total labeling is a bijection α from V (G) ∪ E(G) onto the set of consecutive integers {1, 2, . . . , n+ e}, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d. The vertex-weight of a vertex x is the sum of values α(xy) assigned to all edges xy incident to the v...

A random A yields A+A of size closer to the high end of |A|2. Furthermore, any set A whose members are sufficiently separated has |A+A| close to this high end, and the high end is actually tight for sets A that are geometric progressions, e.g. the first n powers of two. What about the low end of |A|: what is the structure of sets A where |A+A| is close to |A|? We can see that this low-end estim...

A short proof of the following result of Brown and Buhler is given: For any E > 0 there exists n, = no(E) such that if A is an abelian group of odd order IAl > no and BG A with IBI >&IAI. then B must contain three distinct elements X, y, z satisfying x + y = 22.

It has been a long conjecture that there are arbitrarily long arithmetic progressions of primes. As of now, the longest known progression of primes is of length 26 and was discovered by Benoat Perichon and PrimeGrid in April, 2010 ([1]): 43142746595714191+23681770·223092870n for n = 0, 1, · · · , 25. Many mathematicians have spent years trying to prove (or disprove) this conjecture, and even mo...

We present a new numerical abstract domain. This domain automatically detects and proves bounds on the values of program variables. For that purpose, it relates variable values to a clock counter. More precisely, it bounds these values with the i-th iterate of the function [X 7→ α×X+β] applied onM , where i denotes the clock counter and the floating-point numbers α, β, and M are discovered by t...

Article history: Received 22 March 2012 Accepted 12 November 2012 Available online 27 November 2012