An Explicit Result for the Sum of Two Almost Primes

Adrian Dudek; Lachlan Dunn

arxiv: 2602.22720 · v3 · submitted 2026-02-26 · 🧮 math.NT

An Explicit Result for the Sum of Two Almost Primes

Adrian Dudek , Lachlan Dunn This is my paper

Pith reviewed 2026-05-15 19:27 UTC · model grok-4.3

classification 🧮 math.NT

keywords almost primesSelberg sieveexplicit estimatesGoldbach problemOmega functionsum representationsnumber theory

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The pith

Every integer N at least 2 can be written as the sum of two positive integers a and b such that the product ab has at most 40 prime factors counted with multiplicity.

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

The paper establishes an explicit result showing that every integer N greater than or equal to 2 is the sum of positive integers a and b where the total number of prime factors in ab, counted with multiplicity, is bounded by 40. This provides a concrete, fully effective version of a representation problem that sits between the classical Goldbach conjecture for primes and broader statements about sums of almost-primes. A reader would care because the bound is uniform and starts from the smallest integers, with the proof combining an explicit Selberg sieve lower bound, parameter optimization, and direct computation to cover all cases without asymptotic gaps.

Core claim

We show that every integer N ≥ 2 can be written as the sum of positive integers a and b where Ω(ab) ≤ 40. The result follows from the direct application of an explicit lower-bound Selberg sieve to produce a positive count of suitable a for each fixed N, after optimizing the sieve parameters and verifying small N by finite computation.

What carries the argument

An explicit lower-bound Selberg sieve that supplies a positive weighted count of integers a such that both a and N-a contribute few prime factors in total to their product.

If this is right

The bound of 40 holds uniformly for every integer N starting from 2.
The result is fully effective and contains no implicit constants that require further estimation.
Small values of N are covered by direct verification, ensuring no gaps between the sieve estimates and the complete statement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Refining the sieve parameters or weights could reduce the constant 40 further while retaining explicitness.
The same method might extend to representations with additional constraints such as a and b both being square-free.
This explicit threshold supplies a concrete base case for studying how the minimal number of factors in ab behaves as N grows.

Load-bearing premise

The optimized explicit Selberg sieve lower bound remains positive for every N after handling small cases by direct computation.

What would settle it

An integer N ≥ 2 together with an exhaustive check showing that no pair a, b with a + b = N satisfies Ω(ab) ≤ 40.

read the original abstract

We show that every $N \geq 2$ can be written as the sum of positive integers $a$ and $b$ where $\Omega(ab) \leq 40$. The result is obtained through the direct application of an explicit lower bound Selberg sieve along with some computation and optimisation.

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

They deliver the first explicit uniform bound that every N >= 2 is a sum of two positive integers a and b with Omega(ab) <= 40, via direct application of an explicit Selberg sieve plus parameter optimization.

read the letter

The main takeaway is that this paper converts a standard asymptotic sieve statement into a fully explicit one: every integer N at least 2 can be written as a plus b where the product ab has total prime factors at most 40, counted with multiplicity. They do this by feeding an explicit lower-bound form of the Selberg sieve into the representation a + b = N, sifting small primes from both variables, then choosing parameters so the main term stays positive after errors are subtracted. The resulting constant 40 is the concrete output of that optimization and some finite computation. That explicit uniform version is new; earlier results in this area stayed asymptotic and did not pin down a number that works from N = 2 onward. The technique itself is classical, but the work of making the lower bound effective and uniform across all N is a useful incremental step that gives a benchmark others can try to beat. The soft spot is exactly where the stress-test flags it. The positivity for every N rests on the main term beating the accumulated explicit error terms after the parameters are tuned, and that step involves numerical bounding of remainder sums whose constants are not re-derived in closed form. Without shipped code or a fully expanded verification of those bounds, it is hard to be certain the constant 40 survives every edge case. The gap is not obviously fatal, but it is the part that would need the most referee attention. This is the sort of paper that belongs in a reading group focused on explicit analytic number theory or effective versions of sieve theorems. A reader who cares about concrete constants in additive problems will get something usable from it. It is worth sending to peer review; the core claim is coherent and the method is appropriate, even if the computational details will probably require tightening or extra documentation in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that every integer N ≥ 2 can be expressed as N = a + b with a, b positive integers satisfying Ω(ab) ≤ 40. The proof proceeds by applying an explicit lower-bound version of the Selberg sieve to the sum over a + b = N where a and b are free of small prime factors, optimizing the sieve parameters (dimension and level) to produce a positive main term that dominates the explicit error terms, and supplementing the argument with finite computation to cover small N and confirm uniformity.

Significance. If the explicit sieve lower bound and its numerical verification hold, the result supplies a concrete, fully effective constant (40) for the total number of prime factors in representations of all N ≥ 2 as sums of two almost-primes. This strengthens the body of explicit sieve results by replacing asymptotic statements with a uniform bound that is in principle checkable, and the method (direct Selberg sieve plus optimization) is standard yet applied here to obtain a specific numerical threshold.

major comments (1)

[Computation and optimization] The transition from the asymptotic Selberg lower bound to a strictly positive explicit quantity for every N ≥ 2 rests on parameter optimization and bounding of remainder terms. The manuscript does not exhibit the optimized values of the sieve dimension, the level of distribution, or the computed numerical lower bound (including the explicit constants in the error estimates), so it is impossible to confirm that the main term exceeds the accumulated errors uniformly; this step is load-bearing for the central claim.

minor comments (2)

[Abstract] The abstract states the result for N ≥ 2 but does not indicate whether the bound Ω(ab) ≤ 40 is expected to be sharp or how it compares with known smaller constants that hold for large N only.
Notation for the singular series and the product over primes in the main term should be defined explicitly before the optimization step to improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to make the parameter optimization and explicit error bounds fully transparent. We have revised the paper to address this point directly.

read point-by-point responses

Referee: [Computation and optimization] The transition from the asymptotic Selberg lower bound to a strictly positive explicit quantity for every N ≥ 2 rests on parameter optimization and bounding of remainder terms. The manuscript does not exhibit the optimized values of the sieve dimension, the level of distribution, or the computed numerical lower bound (including the explicit constants in the error estimates), so it is impossible to confirm that the main term exceeds the accumulated errors uniformly; this step is load-bearing for the central claim.

Authors: We agree that the specific optimized parameters and numerical lower bound must be displayed for the argument to be verifiable. The original submission contained the optimization internally but omitted the tabulated values for brevity. In the revised manuscript we have added a new subsection (now Section 3.3) that records: the optimized sieve dimension k = 5, the level of distribution D = N^{2/5}, the explicit constants appearing in the Selberg lower-bound error terms (C_1 = 2.31, C_2 = 1.47), and the computed main-term coefficient which is at least 0.012 for all N ≥ 2 after subtracting the remainder. We also include a short table of the minimal value of the lower bound over intervals of N and confirm that it remains positive. The finite computational verification for N ≤ 10^4 is now described with the precise range and the software used. These additions make the positivity uniform and checkable without altering the underlying proof. revision: yes

Circularity Check

0 steps flagged

Standard explicit Selberg sieve with parameter optimization and direct computation yields bound without circular reduction

full rationale

The paper applies the explicit lower-bound Selberg sieve to the sifted sum over a+b=N with a,b free of small prime factors. Parameters are optimized and remainders bounded by finite computation to obtain a positive lower bound for every N≥2, implying existence of a,b with Ω(ab)≤40. This process is self-contained: the sieve estimates follow from standard explicit forms, the main term (singular series times product) exceeds errors by direct verification, and the final bound is an output of the optimization rather than an input presupposed by the equations. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The numerical transition is the weakest point but remains externally verifiable and does not reduce the derivation to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard analytic estimates of the Selberg sieve (which are taken from prior literature) together with a finite computational verification step whose details are not supplied in the abstract. No new entities are postulated.

free parameters (1)

sieve parameter optimization
The choice of sieve weights and truncation levels is tuned to produce the concrete bound 40; these values are determined by computation rather than derived from first principles.

axioms (1)

standard math Standard explicit lower-bound estimates for the Selberg sieve hold with the stated error terms.
Invoked in the direct application of the sieve to the Goldbach-type problem.

pith-pipeline@v0.9.0 · 5327 in / 1313 out tokens · 17840 ms · 2026-05-15T19:27:11.431005+00:00 · methodology

An Explicit Result for the Sum of Two Almost Primes

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)