Representations by Quadratic Forms — Allen Hatcher
Mathematics — Number Theory · representations-by-quadratic-forms
Problem
For discriminant Δ=−24 do the following:
(a) Verify that the class number is 2 and find two quadratic forms Q1 and Q2 of discriminant −24 that are not equivalent.
(b) Draw topographs for Q1 and Q2 showing all values less than 100. (You do not have to repeat parts of the topographs that are symmetric.)
(c) Divide the primes less than 100 into three lists: those represented by Q1, those represented by Q2, and those represented by neither Q1 nor Q2. (No primes are represented by both Q1 and Q2.)
(d) Characterize the primes in the three lists in part (c) in terms of congruence classes mod ∣Δ∣=24.
(e) Characterize the nonprime values in the topograph of Q1 in terms of their factorizations into primes in the lists in part (c), and then do the same thing for Q2. Your answers should be in terms of whether there are an even or an odd number of prime factors from certain of the lists.
(f) Summarize the previous parts by giving a criterion for which numbers n the equation Q1(x,y)=n has an integer solution and likewise for the equation Q2(x,y)=n.