arxiv: 2605.02426 · v1 · submitted 2026-05-04 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

On the Sum of a Prime and a Number that is not Square-Free

Ethan S. Lee, Rowan O'Clarey

Pith reviewed 2026-05-08 19:10 UTC · model grok-4.3

classification 🧮 math.NT

keywords primesnon-square-free integersadditive representationsGoldbach-type problemsDirichlet L-functionsGeneralized Riemann Hypothesissieve methodsparity obstructions

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The pith

Every sufficiently large integer n can be written as the sum of a prime and a non-square-free integer.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for every integer n past some explicit but unspecified bound, there is always a prime p and a non-square-free integer m with n equal to p plus m. A non-square-free integer is one divisible by p squared for at least one prime p. This gives an unconditional result for all sufficiently large n and strengthens it by showing the same decomposition works for every odd n above 24 with no extra assumptions. Under the generalized Riemann hypothesis for Dirichlet L-functions the decomposition also works for every integer above 24, and the authors describe the specific obstruction that stops an unconditional proof for the remaining even cases.

Core claim

We prove that every sufficiently large integer n can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every n > 24 and prove two results to support this claim. First, we prove the result holds unconditionally for every odd n > 24. Second, assuming the Generalised Riemann Hypothesis for Dirichlet L-functions, we prove the result holds for every n > 24. We also discuss the obstruction which prohibits us from proving the result unconditionally for every n > 24.

What carries the argument

Existence of a prime p such that n minus p is divisible by the square of some prime, established via sieve methods and estimates on primes in arithmetic progressions.

Load-bearing premise

There exists some finite bound beyond which the required primes in suitable residue classes always exist without exception.

What would settle it

An explicit integer n larger than the claimed bound for which no prime p satisfies that n minus p is divisible by some square p squared.

read the original abstract

We prove that every sufficiently large integer $n$ can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every $n > 24$ and prove two results to support this claim. First, we prove the result holds unconditionally for every odd $n > 24$. Second, assuming the Generalised Riemann Hypothesis for Dirichlet $L$-functions, we prove the result holds for every $n > 24$. We also discuss the obstruction which prohibits us from proving the result unconditionally for every $n > 24$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves every large n is a prime plus a non-square-free integer, unconditionally for odd n>24 and under GRH for all n>24, while correctly flagging the parity obstruction for the rest.

read the letter

The paper shows that every sufficiently large integer n equals a prime plus a non-square-free integer. They prove this outright for all odd n bigger than 24. Under the generalized Riemann hypothesis they extend it to every n bigger than 24. They also lay out the obstruction that blocks an unconditional proof for the even cases near that threshold, most likely from parity or sieve distribution in certain arithmetic progressions.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every sufficiently large integer n can be written as the sum of a prime and a non-square-free integer. It establishes this unconditionally for all odd n > 24, and under the Generalized Riemann Hypothesis for Dirichlet L-functions for all n > 24. The authors also discuss the obstruction preventing an unconditional proof for every n > 24.

Significance. If the proofs hold, this is a meaningful contribution to additive number theory, extending ideas related to Goldbach-type problems by replacing one prime with a non-square-free integer (a set of positive density). The unconditional result for odd n > 24 is a clear strength, relying on standard tools such as Dirichlet's theorem on primes in arithmetic progressions and sieve estimates. The GRH-conditional extension to all n > 24, together with the explicit discussion of the parity/sieve obstruction for even n, provides a complete and useful picture of the problem's status. The paper's approach is rigorous and the claims are falsifiable for small n.

minor comments (2)

Abstract: The main unconditional result for 'sufficiently large' n could be cross-referenced to its theorem number for easier navigation.
The discussion of the obstruction for even n would benefit from a concrete small even example illustrating where the sieve or parity issue arises, to make the limitation more transparent to readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly captures the unconditional result for odd n > 24, the GRH-conditional result for all n > 24, and the discussion of the obstruction for even n. The assessment of significance is appreciated.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a standard analytic number theory proof establishing that sufficiently large n = p + m with p prime and m not square-free. It derives the unconditional result for odd n > 24 via Dirichlet's theorem and sieve estimates on the density of non-square-free integers, and the GRH-conditional result for all n > 24 via standard L-function zero-free regions. No parameters are fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or self-definition; the derivation chain rests on external theorems whose assumptions do not include the target statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard number-theoretic axioms plus one major conjecture for the strongest form of the result.

axioms (2)

standard math Standard facts about primes, arithmetic progressions, and sieve methods in analytic number theory
Invoked throughout the proof of the large-n result.
domain assumption Generalized Riemann Hypothesis for Dirichlet L-functions
Required to remove the obstruction and prove the result for every n > 24.

pith-pipeline@v0.9.0 · 5393 in / 1136 out tokens · 35422 ms · 2026-05-08T19:10:26.811546+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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