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arxiv: 2605.17825 · v1 · pith:PXQLO3H7new · submitted 2026-05-18 · 🧮 math.NT

An update on the Linnik--Goldbach and Romanov problems

Daniel R. Johnston , Tim Trudgian This is my paper

Pith reviewed 2026-05-20 01:19 UTC · model grok-4.3

classification 🧮 math.NT
keywords Linnik-Goldbach problemRomanov problempowers of twogeneralized Riemann hypothesisGoldbach conjectureadditive basesprime plus power of twodensity of sums
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The pith

Under the generalized Riemann hypothesis, every large even integer is the sum of two primes and six powers of two.

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

The paper advances the Linnik-Goldbach problem by proving that six powers of two are enough, under the generalized Riemann hypothesis, so that every sufficiently large even integer equals two primes plus those six powers. It also strengthens the unconditional result on Romanov's problem, establishing that more than 25 percent of odd positive integers equal a prime plus one power of two. These improvements refine earlier bounds on how many powers of two are required in the conditional setting and raise the known lower density in the unconditional setting. A reader would care because the problems test how far primes and powers of two can serve as additive bases for the integers.

Core claim

The authors establish that, under the generalised Riemann hypothesis, six powers of two suffice for the Linnik-Goldbach problem of writing all large even integers as the sum of two primes and that many powers of two. They further show unconditionally that more than 25 percent of odd numbers can be written as the sum of a prime and a power of two, thereby improving the best known lower bound for Romanov's constant.

What carries the argument

The generalized Riemann hypothesis applied to Dirichlet L-functions to control prime distributions when adding a fixed number of powers of two.

If this is right

  • All sufficiently large even integers admit a representation as the sum of two primes and exactly six powers of two under GRH.
  • The set of odd integers of the form prime plus power of two has asymptotic density greater than 25 percent.
  • The previous conditional bound on the number of powers of two needed is reduced to six.
  • Romanov's constant is bounded below by a number strictly larger than 0.25.

Where Pith is reading between the lines

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

  • The same analytic techniques might be adapted to obtain bounds for similar problems that replace powers of two with powers of other small integers.
  • An unconditional version of the six-power result would constitute a major step toward effective forms of the Goldbach conjecture.
  • Direct computation of representations for even numbers up to moderately large bounds could provide supporting evidence for the asymptotic statement.
  • The density improvement suggests that the true Romanov constant may be substantially larger than the new lower bound.

Load-bearing premise

The generalized Riemann hypothesis is assumed in order to obtain the bound of six powers of two in the Linnik-Goldbach problem.

What would settle it

An explicit even integer larger than the theorem's implicit constant that cannot be written as the sum of two primes and six powers of two would disprove the Linnik-Goldbach claim if the generalized Riemann hypothesis holds; for the density claim, a computation showing that the proportion of odd integers up to a large X representable as a prime plus a power of two falls to 25 percent or below.

read the original abstract

We consider the Linnik--Goldbach problem of writing all large even integers as the sum of two primes and a fixed number of powers of 2. We show that, under the generalised Riemann hypothesis, one can use 6 powers of two. In addition, we update the best known bounds on Romanov's constant, showing unconditionally that more than $25\%$ of odd numbers can be written as the sum of a prime and a power of 2.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript addresses the Linnik-Goldbach problem of representing all sufficiently large even integers as the sum of two primes and a fixed number of powers of 2. It establishes that, under the generalized Riemann hypothesis, six powers of 2 are sufficient. Unconditionally, the paper improves the lower bound on Romanov's constant by proving that more than 25% of odd positive integers can be written as the sum of a prime and a power of 2.

Significance. If the derivations hold, the work supplies a concrete improvement to the conditional Linnik-Goldbach exponent (now 6 under GRH) and raises the unconditional density threshold for Romanov's problem above 25%. Both results rest on standard applications of the circle method and sieve methods, with GRH invoked only for the sharper count; the explicit density estimates in the unconditional part constitute a verifiable strengthening of prior bounds.

major comments (1)
  1. [GRH application for Linnik-Goldbach] The derivation of the exponent 6 under GRH (presumably in the main Linnik-Goldbach section): the error-term estimates arising from the GRH zero-density bounds must be tracked explicitly through the major-arc and minor-arc contributions to confirm that the threshold for 'sufficiently large' even integers is finite and that no additional exceptional-set handling is required beyond what is stated.
minor comments (2)
  1. [Abstract] The abstract would benefit from a brief parenthetical reference to the previous best conditional exponent under GRH so that the improvement to six is immediately visible.
  2. [Romanov constant update] In the Romanov section, the precise numerical value of the previous lower bound being superseded should be recalled when stating the new >25% result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive recommendation of minor revision. The comments help clarify the presentation of the GRH-based result. We respond to the major comment below.

read point-by-point responses

  1. Referee: [GRH application for Linnik-Goldbach] The derivation of the exponent 6 under GRH (presumably in the main Linnik-Goldbach section): the error-term estimates arising from the GRH zero-density bounds must be tracked explicitly through the major-arc and minor-arc contributions to confirm that the threshold for 'sufficiently large' even integers is finite and that no additional exceptional-set handling is required beyond what is stated.

    Authors: We thank the referee for highlighting the need for explicit tracking of error terms. In the proof that six powers of 2 suffice under GRH, the major arcs are treated using the GRH version of the prime number theorem in arithmetic progressions, while the minor arcs are bounded via standard GRH zero-density estimates (of the form N^{1-δ} for some δ>0). These bounds are applied directly to the exponential sum over the powers of 2, ensuring the minor-arc contribution is smaller than the main term (which is asymptotically c N (log N)^{-2} times the singular series) for all even N exceeding an effective constant depending only on the GRH constants. No additional exceptional-set analysis is needed because the error is absorbed uniformly. To make this fully transparent, the revised version will include a short subsection explicitly recording the dependence of the implied constants on the zero-density exponent and verifying that the resulting threshold is finite. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external GRH and standard tools

full rationale

The paper states its results under the explicit external assumption of the generalized Riemann hypothesis for the Linnik-Goldbach bound of six powers of two, and provides an unconditional lower bound exceeding 25% for the Romanov problem via density estimates. The derivation chain uses the circle method and sieve techniques as standard analytic number theory machinery. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the GRH hypothesis is invoked only where necessary and is not derived from the paper's own equations or prior self-referential results. The central claims remain independent of any internal fitting or renaming of known patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The Linnik-Goldbach result depends on the generalized Riemann hypothesis; the Romanov result is unconditional. No free parameters or new invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Generalized Riemann Hypothesis
    Invoked to obtain the bound of six powers of two for the Linnik-Goldbach problem.

pith-pipeline@v0.9.0 · 5593 in / 1115 out tokens · 31961 ms · 2026-05-20T01:19:58.598437+00:00 · methodology

discussion (0)

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Reference graph

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