Arithmetic Progressions of Four Squares

Suppose a, b, c, and d are rational numbers such that a2, b2, c2, and d2 form an arithmetic progression: the differences b2−a2, c2−b2, and d2−c2 are equal. One possibility is that the arithmetic progression is constant: a2, a2, a2, a2. Are there arithmetic progressions of four rational squares which are not constant? This question was first raised by Fermat in 1640. There are no such progressions with small rational squares, but that doesn’t preclude the possibility of a rational solution altogether. After all, the smallest positive integer solution to x2 − 61y2 = 1 is (x, y) = (1766319049, 226153980). We will show how to turn 4-tuples of numbers (not all 0) whose squares form an arithmetic progression into points on the elliptic curve