Arithmetic Progressions with Square Entries

We study properties of arithmetic progressions consisting of three squares; in particular, how one arithmetic progression generates infinitely many others, by means of explicit formulas as well as a matrix method. This suggests an equivalence relation could be defined on the arithmetic progressions, which lead to interesting problems for further study. The purpose of this paper is to investigate ordered triples of integers whose squares form an arithmetic progression. In other words, we want to study (a, b, c), where a, b, c are integers that satisfy b − a = c − b, or equivalently, 2b = a + c. (1) We call such an ordered triple an arithmetic progression triple, or simply an apt. Theorem 1: The ordered triple (a, b, c) is an apt if and only if it satisfies equation (1). Obviously, we have an apt if its entries have the same absolute value. Consequently, for n 6= ±1, we call the ordered triple (±n,±n,±n), for any combination of signs, the trivial apts. Examples of nontrivial apts include (1,−1,−1), (1, 5, 7), and (−7, 13, 17). Solving equation (1) is a rather standard exercise. A proof of the next result can be found in, for example, [6, pages 305 and 343]. It can also be derived from a more general result regarding solutions of ax + bxy + cy = ez; see, for example, [2, Theorem 42]. Theorem 2: Let ρ be the greatest common divisor of a, b, c, then the solutions of the Diophantine equation 2b = a + c are of the form a = ±ρ(m − n − 2mn), b = ±ρ(m + n), c = ±ρ(m − n + 2mn), (2) for any integers m and n. In light of Theorem 2, we call the apt (a, b, c) a primitive arithmetic progression triple, or simply a papt, if ρ = 1. Observe that • If (a, b, c) is a papt, then so is (c, b, a). • If (a, b, c) is a papt, then so are (±a,±b,±c) for any combination of signs. • Consequently, with the exception of (±1,±1,±1), finding one papt would immediately lead to fifteen other papts, which vary only in signs and order. The ordered triple (±1,±1,±1), however, leads to only seven other papts.