Distance Between Arithmetic Progressions and Perfect Squares

Abstract

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.Many questions in number theory can be phrased as the study of the "distances" between two sequences of numbers. For instance, we have the famous conjecture that there are infinitely many primes of the form n 2 + 1. This can be interpreted as saying that the sequence of prime numbers and the sequence of perfect squares can get within unit distance from one another infinitely often. Another example is the conjecture of M. Hall, Jr. stating that if x 2 = y 3 , then |x 2 − y 3 | ≫ x 1/2 . This means that there are significant gaps between the sequence of perfect squares and the sequence of perfect cubes (apart from the sequence of sixth powers). In this paper, we are going to consider the distance between two simplest arithmetic sequences, namely arithmetic progressions and perfect squares.More specifically, given integers a ≥ 0, d ≥ 1 and N ≥ 1, we consider the arithmetic progression A = A a,d,N := {a, a + d, a + 2d, ..., a + N d}.We are interested in how close the terms of the above arithmetic progression can get to a perfect square 0 2 , 1 2 , 2 2 , 3 2 , .... More precisely, we let δ = δ a,d,N := min 0≤n≤N m∈Z