arxiv: 2605.14369 · v1 · pith:GMUCINFVnew · submitted 2026-05-14 · 🧮 math.NT

A density version of quaternary Goldbach problem

Xiaoyang Hu , Meng Gao This is my paper

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keywords deltaunderlineprimesdenotedensitymathcalbestcondition

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The pith

If four subsets of primes satisfy underline delta(P1) + underline delta(P2) > 1 and underline delta(P3) + underline delta(P4) > 1, then every sufficiently large even n is the sum of one prime from each subset, and the bound is sharp.

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

The classical Goldbach conjecture says every even number bigger than 2 is the sum of two primes. Here the authors look at four groups of primes that are not necessarily all the primes, but each group is dense enough in the primes. The lower density measures how much of the primes sit in each group. When two pairs of these groups each cover more than half the primes in a density sense, their sums fill all large even numbers. The paper shows this threshold cannot be lowered without creating counterexamples.

Core claim

Suppose that P1, P2, P3, P4 are four subsets of primes with underline delta(P1)+underline delta(P2)>1 and underline delta(P3)+underline delta(P4)>1. Then for every sufficiently large even integer n, there exist primes pi in Pi (i=1,2,3,4) such that n=p1+p2+p3+p4. The condition is the best possible.

Load-bearing premise

The proof assumes that the density conditions are sufficient to apply some form of the circle method or sieve to control the singular series and minor arcs for the quaternary sum; if the analytic estimates fail at the stated density threshold, the representation claim would not hold.

read the original abstract

Let $\mathcal{P}$ denote the set of all primes, and let $\underline\delta(P)$ denote the relative lower density of a subset $P$ in $\mathcal{P}$. Suppose that $P_1, P_2, P_3, P_4$ are four subsets of primes with $\underline\delta(P_1)+\underline\delta(P_2)>1$ and $ \underline\delta(P_3)+\underline\delta(P_4)>1.$ Then for every sufficiently large even integer $n$, there exist primes $p_i \in P_i$ $(i=1,2,3,4)$ such that $n=p_1+p_2+p_3+p_4$. The condition is the best possible.

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard analytic number theory tools (circle method, sieve estimates for primes) whose validity at the given density level is the main technical step; no new free parameters or invented entities are introduced in the statement.

axioms (1)

standard math Standard properties of the von Mangoldt function and the distribution of primes in arithmetic progressions
Invoked implicitly to control the major arcs and singular series in the representation.

pith-pipeline@v0.9.0 · 5411 in / 1219 out tokens · 56028 ms · 2026-05-15T02:18:37.663348+00:00 · methodology

discussion (0)

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Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

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