On the Sum of the Square of a Prime and a Square-Free Number

TL;DR

This paper proves that all integers greater than or equal to 10, not congruent to 1 mod 4, can be expressed as the sum of a prime square and a square-free number, extending Erdős's theorem with explicit bounds.

Contribution

It provides explicit constructions and bounds for representing integers as the sum of a prime square and a square-free number, building on and extending previous theoretical results.

Findings

All integers n ≥ 10, n ≠ 1 mod 4, can be expressed as such sums.

New explicit bounds for primes in arithmetic progressions are established.

Numerical computations under GRH extend previous explicit bounds.

Abstract

We prove that every integer $n \geq 10$ such that $n \not\equiv 1 \text{mod} 4$ can be written as the sum of the square of a prime and a square-free number. This makes explicit a theorem of Erd\H{o}s that every sufficiently large integer of this type may be written in such a way. Our proof requires us to construct new explicit results for primes in arithmetic progressions. As such, we use the second author's numerical computation regarding GRH to extend the explicit bounds of Ramar\'e-Rumely.