lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses multiple nested inductions, which result in extremely weak bounds for N{h,k). We shall define Bk to...

Given an integer q ≥ 2 and a number θ ∈ (0, 1], consider the collection of all subsets of Zq := Z/qZ having at least θq elements. Among the sets in this collection, suppose S is any one having the minimal number of three-term arithmetic progressions, where in our terminology a three-term arithmetic progression is a triple (x, y, z) ∈ S3 satisfying x + y ≡ 2z (mod q). Note that this includes tri...

is a (nonconstant) arithmetic progression of positive integers. We consider a general binary quadratic form ax2 + bxy + cy' ( a , b , c E Z ) and ask the question "Can the form ax' + hxy + ry' represen1 every inleger in 1he arithmetic progression kNo + 1 for any natural numbers k and l?" In a sampling of books containing a discussion of binary quadratic forms [2]-[9], we did not find this qustl...

Let N denote the set of all nonnegative integers. Let k ≥ 3 be an integer and A0 = {a1, . . . , at} (a1 < . . . < at) be a nonnegative set which does not contain an arithmetic progression of length k. We denote A = {a1, a2, . . . } defined by the following greedy algorithm: if l ≥ t and a1, . . . , al have already been defined, then al+1 is the smallest integer a > al such that {a1, . . . , al}...

A celebrated theorem of Erdős and Selfridge [14] states that the product of consecutive positive integers is never a perfect power. A more recent and equally appealing result is one of Darmon and Merel [11] who proved an old conjecture of Dénes to the effect that there do not exist three consecutive nth powers in arithmetic progression, provided n 3. One common generalization of these problems ...

We investigate a method of ordering pixels (the elements of a rectangular matrix) based on an arithmetic progression with wrap-around (modular arithmetic). For appropriate choices of the progression's parameters based on a generalization of Fibonacci numbers and the golden mean, we nd equidis-tributed collections of pixels formed by subin-tervals of the pixel progression or \shuue." We illustra...

We consider those positive integers that are not representable as linear combinations of terms of a generalized arithmetic progression with nonnegative integer coefficients. To do this, we make use of the numerical semigroup generated by a generalized arithmetic progression. The number of integers nonrepresentable by such a numerical semigroup is determined as well as that of its dual. In addit...

We consider arithmetic progressions on Pellian equations x2 − d y2 = m, i.e. integral solutions such that the y-coordinates are in arithmetic progression. We construct explicit infinite families of d,m for which there exists a five-term arithmetic progression in the y-coordinate, and we prove the existence of infinitely many pairs d,m parametrized by the points of an elliptic curve of positive ...

We present a method of ordering pixels (the elements of a rectangular matrix) based on an arithmetic progression with wrap-around (modular arithmetic). For appropriate choices of the progression’s parameters based on a generalization of Fibonacci numbers and the golden mean, we achieve uniformly distributed collections of pixels formed by intervals of the pixel progression or “shuffle.” We illu...