Principal forms X^2 + nY^2 representing many integers

Details Principal forms X^2 + nY^2 representing many integers Principal forms X^2 + nY^2 representing many integers

2010-03-04

arxiv.org/abs/1003.1094v2

Mathematics

Number Theory

10 pages, title has been changed, Sections 2 and 3 are new, to appear in Abh. Math. Sem. Univ. Hamburg

Scientific paper

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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