ABSTRACT
Except for the last section, this chapter has little to do with sums of squares of integers. It proves a famous theorem of Szemere´di that says that every subset of the positive integers with positive (upper) asymptotic density contains arbitrarily long arithmetic progressions. Three squares x2, y2, z2 form an arithmetic progression if and only if y2 − x2 = z2 − y2, that is, if and only if x2 + z2 = 2y2. Therefore, an arithmetic progression of three squares is equivalent to the representation of twice a square as the sum of two squares. Exercises at the end of this chapter make additional connections between arithmetic progressions and sums of squares. A further connection between this chapter and the rest of the book is a proof technique (in Section 7.3) which was also used in Section 6.2.
