Five peculiar theorems on simultaneous representation of primes by quadratic forms

Text. It is a theorem of Kaplansky that a prime p equivalent to 1 (mod 16) is representable by both or none of x(2) + 32y(2) and x(2) + 64y(2), whereas a prime p equivalent to 9 (mod 16) 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 equivalent to 1 (mod 20) is representable by both or none of x(2) + 20y(2) and x(2) + 100y(2), whereas a prime p equivalent to 9 (mod 20) 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 Delta such that the class group l(Delta) has exponent 4. Video. For a video summary of this paper. please visit http://www.youtube.com/watch?v=l_yRq0oqKx4. (C) 2008 Elsevier Inc. All rights reserved.