Density of the union of positive diagonal binary quadratic forms

TL;DR

This paper demonstrates that a positive proportion of integers less than a large number X can be represented by quadratic forms involving a small set of primes, with the set size growing logarithmically with X.

Contribution

It introduces a method to select a small prime subset ensuring widespread representability of integers by specific quadratic forms.

Findings

A small prime subset of size O(log X) suffices for broad integer representation.

A positive proportion of integers less than X are representable by forms x^2 + p y^2 with p in the subset.

The density of such representations is significant for large X.

Abstract

Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least one of $p \in S$.