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

TL;DR

This paper proves that the constant in the asymptotic formula for counting integers represented by the quadratic form X^2 + nY^2 becomes unbounded as n varies over positive integers with a fixed number of prime divisors, confirming numerical observations.

Contribution

It establishes the unboundedness of the main constant for the quadratic form X^2 + nY^2 as n varies with a fixed number of prime factors, advancing understanding of representation counts.

Findings

The main constant in the asymptotic formula is unbounded for fixed prime divisor count.

Numerical observations by Shanks and Schmid are theoretically confirmed.

The result applies to integers represented by X^2 + nY^2 with fixed prime divisors.

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.