Five peculiar theorems on simultaneous representation of primes by quadratic forms

It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p ≡ 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 ≡ 1 (mod 20) is representable by both or none of x2 + 20y2 and x2 + 100y2, whereas a prime p ≡ 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. The latter argument relies on the (highly plausible) conjecture that there are no negative discriminants ∆ other than the 485 known ones such that the class group C (∆) has exponent 4. The methods are purely classical. Consider a negative integer ∆ ≡ 0, 1 (mod 4) and recall that the principal form F (x, y) of discriminant ∆ is x− ∆ 4 y or x +xy− ∆−1 4 y according to ∆’s parity. It is well known that the prime numbers representable by F (x, y) are describable by congruence conditions if and only if each genus of forms of discriminant ∆ consists of a single class, or – equivalently – the class group C (∆) is either trivial or has exponent 2 (see e.g. [2, p. 62]). The determination of the negative discriminants with one class per genus is a famous problem in number theory. Gauss, who considered only forms of type ax + 2bxy + cy, found 65 such discriminants −4n and showed that they correspond to Euler’s idoneal or convenient numbers n. Dickson [3] compiled a ∗Institut for Matematiske Fag, Københavns Universitet, Universitetsparken 5, DK-2100, Denmark. E-mail: [email protected]. In what follows, all forms are binary, quadratic, primitive, and positive definite.