Five peculiar theorems on simultaneous representation of primes by quadratic forms

TextIt is a theorem of Kaplansky that a prime p≡1(mod16) is representable by both or none of x2+32y2x2+32y2 and x2+64y2x2+64y2, whereas a prime p≡9(mod16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved. As an example, one theorem states that a prime p≡1(mod20) is representable by both or none of x2+20y2x2+20y2 and x2+100y2x2+100y2, whereas a prime p≡9(mod20) is representable by exactly one of these forms. A heuristic argument is given why there are no other results of the same kind. This argument relies on the (plausible) conjecture that there are exactly 485 negative discriminants Δ   such that the class group C(Δ)C(Δ) has exponent 4.VideoFor a video summary of this paper, please visit http://www.youtube.com/watch?v=l_yRq0oqKx4.