(From attached notes) The type of question we will examine has it roots in a famous result of Heath-Brown on the statistics of 2-Selmer ranks of a specific family of CM elliptic curves over Q related to the congruent number problem. This is the family E_D: Dy^2 = x^3-x for positive square-free integers D. The arithmetic of this family answers the question of whether or not D can be the common difference of an arithmetic progressions of squares of rational numbers.