Arithmetic progressions of squares, cubes and $n$-th powers

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...

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 progressio...

Sums of Squares, Cubes, and Higher Powers

Arithmetic progressions of four squares over quadratic fields

Let d be a squarefree integer. Does there exist four squares in arithmetic progression over Q( √ d )? We shall give a partial answer to this question, depending on the value of d. In the affirmative case, we construct explicit arithmetic progressions consisting of four squares over Q( √ d ).

Lattice Points on Circles, Squares in Arithmetic Progressions and Sumsets of Squares

Let σ(k) denote the maximum of the number of squares in a+b, . . . , a+kb as we vary over positive integers a and b. Erdős conjectured that σ(k) = o(k) which Szemerédi [30] elegantly proved as follows: If there are more than δk squares amongst the integers a+b, . . . , a+kb (where k is sufficiently large) then there exists four indices 1 ≤ i1 < i2 < i3 < i4 ≤ k in arithmetic progression such th...