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Principal forms X2+nY2 representing many integers
Authors:David Brink  Pieter Moree  Robert Osburn
Affiliation:1.School of Mathematical Sciences,University College Dublin,Dublin 4,Ireland;2.Max-Planck-Institut für Mathematik,Bonn,Germany
Abstract:In 1966, Shanks and Schmid investigated the asymptotic behavior of the number of positive integers less than or equal to x which are represented by the quadratic form X 2+nY 2. Based on some numerical computations, they observed that the constant occurring in the main term appears to be the largest for n=2. In this paper, we prove that in fact this constant is unbounded as n runs through positive integers with a fixed number of prime divisors.
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