Arithmetic Progressions of Three Squares

In this list there is an arithmetic progression: 1, 25, 49 (common difference 24). If we search further along, another arithmetic progression of squares is found: 289, 625, 961 (common difference 336). Yet another is 529, 1369, 2209 (common difference 840). How can these examples, and all others, be found? In Section 2 we will use plane geometry to describe the 3-term arithmetic progressions of (nonzero) perfect squares in terms of rational points on the circle x2+y2 = 2. Since a2, b2, c2 is an arithmetic progression if and only if (a/d)2, (b/d)2, (c/d)2 is an arithmetic progression, where d 6= 0, there is not much difference between integral and rational arithmetic progressions of three squares, and in Section 3 we will describe 3-term arithmetic progressions of rational squares with a fixed common difference in terms of rational points on elliptic curves (Corollary 3.6). In the appendix, the link between elliptic curves and arithmetic progressions with a fixed common difference is revisited using projective geometry.