Finiteness theorems for universal sums of squares of almost primes

TL;DR

This paper investigates quadratic forms that are universal over inputs with a limited number of prime factors, proving finiteness theorems similar to the 15 theorem for sums of squares of almost primes.

Contribution

It establishes new finiteness theorems for quadratic forms that are universal over almost prime inputs, extending classical results to a broader class of numbers.

Findings

Proved finiteness theorems for universal sums of squares of almost primes.

Extended classical universality results to almost prime inputs.

Identified conditions under which quadratic forms are universal for almost primes.

Abstract

In this paper we study quadratic forms which are universal when restricted to almost prime inputs, establishing finiteness theorems akin to the Conway--Schneeberger 15 theorem.