Representations by Quadratic Forms — Allen Hatcher
Mathematics — Number Theory · representations-by-quadratic-forms
Problem
Consider a discriminant Δ=q2, q>0, corresponding to 0-hyperbolic forms. Using the description of the topographs of such forms obtained in the previous chapter, show:
(a) Every number is represented by at least one form of discriminant Δ, so in particular all primes are represented.
(b) The primes represented by a given form of discriminant Δ are exactly the primes in certain congruence classes mod q (and hence also mod Δ).
(c) For q=1,2,7, and 15 determine the class number for discriminant Δ=q2 and find which primes are represented by the forms in each equivalence class.