Quadratic residue patterns, algebraic curves and a K3 surface

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive -tuples of quadratic residues, studying corresponding algebraic curves and their jacobians, which happen to be products of jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.