On Triangles with Sides That Form an Arithmetic Progression

Heron Quadrilaterals with Sides in Arithmetic or Geometric Progression

We study triangles and cyclic quadrilaterals which have rational area and whose sides form geometric or arithmetic progressions. A complete characterization is given for the infinite family of triangles with sides in arithmetic progression. We show that there are no triangles with sides in geometric progression. We also show that apart from the square there are no cyclic quadrilaterals whose si...

On Triples in Arithmetic Progression

On the Second Moment for Primes in an Arithmetic Progression

Abstract. Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together w...

Dirichlet’s Theorem on Primes in an Arithmetic Progression

Our goal is to prove the following theorem: Dirichlet’s Theorem: For any coprime a, b ∈ Z, there are infinitely many primes p such that p ≡ a (mod b). Although the statement of the theorem involves only integers, the simplest proof requires the use of complex numbers and Dirichlet L-series. Most of this paper will therefore be devoted to proving some basic properties of characters and L-series,...

Positional Number Systems with Digits Forming an Arithmetic Progression

Abstract. A novel digit system that arises in a natural way in a graph-theoretical problem is studied. It is defined by a set of positive digits forming an arithmetic progression and, necessarily, a complete residue system modulo the base b. Since this is not enough to guarantee existence of a digital representation, the most significant digit is allowed to come from an extended set. We provide...