I. Let Q(N; q; a) denote the number of squares in the arithmetic progression qn+a; n = 1; 2; ; N; and let Q(N) be the maximum of Q(N; q; a) over all non-trivial arithmetic progressions qn + a. It seems to be remarkably diicult to obtain non-trivial upper bounds for Q(N). There are currently two proofs known of the weak bound Q(N) = o(N) (which is an old conjecture of Erdd os) and both are far f...

Prime numbers have fascinated people since ancient times. Since the last century, their study has acquired importance also on account of the crucial role played by them in cryptography and other related areas. One of the problems about primes which has intrigued mathematicians is whether it is possible to have long strings of primes with the successive primes differing by a fixed number, namely...

Let N(k) = 2k/2k3/2f(k) and N(k) = 2k/2k1/2 g(k) where f(k) → ∞ and g(k) → 0 arbitrarily slowly as k → ∞. We show that the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic kterm arithmetic progression approaches 0, as k → ...

We look at arithmetic progressions on elliptic curves known as Huff curves. By an arithmetic progression on an elliptic curve, we mean that either the x or y-coordinates of a sequence of rational points on the curve form an arithmetic progression. Previous work has found arithmetic progressions on Weierstrass curves, quartic curves, Edwards curves, and genus 2 curves. We find an infinite number...

In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with b...