Representations by Quadratic Forms — Allen Hatcher
Mathematics — Number Theory · representations-by-quadratic-forms
Problem
For the form Q(x,y)=x2+xy−y2 do the following things:
(a) Draw enough of the topograph to show all the values less than 100 that occur in the topograph. This form is hyperbolic and it takes the same negative values as positive values, so you need not draw all the negative values.
(b) Make a list of the primes less than 100 that occur in the topograph, and a list of the primes less than 100 that do not occur.
(c) Characterize the primes in the two lists in part (b) in terms of congruence classes mod ∣Δ∣ where Δ is the discriminant of Q.
(d) Characterize the nonprime values in the topograph in terms of their factorizations into primes in the lists in part (b).
(e) Summarize the previous parts by giving a simple criterion for determining the numbers n such that Q(x,y)=n has an integer solution (x,y), primitive or not. The criterion should say something like Q(x,y)=n is solvable if and only if n=m2p1⋯pk where each pi is a prime such that …
(e) Check that all forms having the same discriminant as Q are equivalent to Q.