It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it is necessary that a prescribed set of at least 1254 numbers between the first and last prime all be composi...

In this paper we study the problem of labeling the edges of a graph with positive integers such that the sequence of the sums of incident edges of each vertex makes a finite arithmetic progression. We give conditions for paths, cycles, and bipartite graphs to have such a labeling. We then address the opposite problem of finding an edge labeled graph for a given finite arithmetic progression. We...

This paper is motivated by a recent result of Wolf [12] on the minimum number of monochromatic 4-term arithmetic progressions (4-APs, for short) in Zp, where p is a prime number. Wolf proved that there is a 2-coloring of Zp with 0.000386% fewer monochromatic 4-APs than random 2-colorings; the proof is probabilistic and non-constructive. In this paper, we present an explicit and simple construct...

In 1967 the first set of 6 consecutive primes in arithmetic progression was found. In 1995 the first set of 7 consecutive primes in arithmetic progression was found. Between November, 1997 and March, 1998, we succeeded in finding sets of 8, 9 and 10 consecutive primes in arithmetic progression. This was made possible because of the increase in computer capability and availability, and the abili...

Let G(k;r) denote the smallest positive integer g such that if 1 = a1;a2; : : : ;ag is a strictly increasing sequence of integers with bounded gaps a j+1 a j r, 1 j g 1, then fa1;a2; : : : ;agg contains a k-term arithmetic progression. It is shown that G(k;2) > q k 1 2 4 3 k 1 2 , G(k;3) > 2 k 2 ek (1+ o(1)), G(k;2r 1)> r k 2 ek (1+o(1)), r 2. For positive integers k, r, the van der Waerden num...