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arxiv: 2605.21876 · v1 · pith:UOHTIG3Mnew · submitted 2026-05-21 · 🧮 math.NT

The reverse Goldbach problem and a refined Zsiflaw--Legeis theorem

Michael Harm , Daniel R. Johnston This is my paper

Pith reviewed 2026-05-22 06:14 UTC · model grok-4.3

classification 🧮 math.NT
keywords reversed primesGoldbach conjectureadditive number theoryarithmetic progressionscircle methodZsiflaw-Legeis theoremsquare-free numbers
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The pith

Large odd integers can be written as the sum of one prime and two reversed primes.

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

The paper establishes several new additive representations for large integers involving reversed primes, which are primes written backwards in a fixed base. It shows that every large odd integer is the sum of a prime and two reversed primes, and also as two primes plus one reversed prime. Additionally, almost all even integers are sums of a prime and a reversed prime, and all large integers are a reversed prime plus a square-free number. A sympathetic reader would care because these results extend classical Goldbach-type problems to a setting where digit reversal is involved, suggesting that such reversals do not disrupt the additive structure of primes in significant ways. The proofs rely on analytic number theory tools to handle the distribution of these reversed primes.

Core claim

Using the Hardy-Littlewood circle method and a novel refinement of the Zsiflaw-Legeis theorem that applies to reversed primes in arithmetic progressions without fixing the digit length, the authors prove that every sufficiently large odd integer N can be expressed as N = p1 + ←p2 + ←p3 where p1 is prime and ←p2, ←p3 are reversed primes, along with the other stated representations.

What carries the argument

A refined version of the Zsiflaw-Legeis theorem on the distribution of reversed primes in arithmetic progressions, which allows variable digit lengths and is used with the circle method to obtain asymptotic formulas for the representations.

Load-bearing premise

The refined Zsiflaw-Legeis theorem holds for the distribution of reversed primes in arithmetic progressions without needing to fix the number of digits.

What would settle it

A counterexample would be a large odd integer that cannot be expressed as the sum of a prime and two reversed primes, or a violation of the asymptotic distribution given by the refined Zsiflaw-Legeis theorem.

read the original abstract

We prove new results on the additive theory of reversed primes $\overleftarrow{p}$; that is, primes $p$ which are written backwards in a fixed base $b\geq 2$. In particular, we study a variant of Goldbach's conjecture, looking at representations of integers as the sum of primes and reversed primes. We show that: (1) Every large odd integer is the sum of a prime and two reversed primes ($N=p_1+\overleftarrow{p_2}+\overleftarrow{p_3}$). (2) Every large odd integer is the sum of two primes and a reversed prime ($N=p_1+p_2+\overleftarrow{p_3}$). (3) Almost all even integers are the sum of a prime and a reversed prime ($N=p_1+\overleftarrow{p_2}$). (4) All large integers are the sum of a reversed prime and a square-free number ($N=\overleftarrow{p}+\eta$, $\mu^2(\eta)=1$). To obtain our results, along with associated asymptotics, we apply the Hardy--Littlewood circle method and a novel refinement of the ``Zsiflaw--Legeis" theorem on the distribution of reversed primes in arithmetic progressions. Notably, our variant of the Zsiflaw--Legeis theorem does not require one to fix the digit length unlike previous versions.

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

2 major / 2 minor

Summary. The manuscript proves four results on additive representations involving reversed primes (primes written backwards in base b ≥ 2): every large odd integer is the sum of one prime and two reversed primes; every large odd integer is the sum of two primes and one reversed prime; almost all even integers are the sum of a prime and a reversed prime; and every large integer is the sum of a reversed prime and a square-free integer. The proofs rely on the Hardy–Littlewood circle method together with a novel refinement of the Zsiflaw–Legeis theorem asserting that reversed primes are equidistributed in arithmetic progressions uniformly without fixing the number of digits.

Significance. If the uniformity in the refined Zsiflaw–Legeis theorem can be established with adequate error-term control, the results would extend classical Goldbach-type theorems to the reversed-prime setting and supply associated asymptotics. The refined distribution theorem itself would be of independent interest for analytic number theory applications involving digit-reversed objects.

major comments (2)
  1. [§3] §3 (refined Zsiflaw–Legeis theorem): the error term is stated without an explicit uniform bound in the digit length d when d grows with N (up to ~log_b N). This uniformity is required for the major-arc approximation and minor-arc bound in the circle-method applications to claims (1)–(3); without it the main-term dominance is not guaranteed.
  2. [§5] §5 (proof of Theorem 1.1): the singular series for the representation N = p + ←p2 + ←p3 is asserted to be asymptotically positive, but the argument does not address the possible dependence on the digit lengths of the reversed primes when they are allowed to vary.
minor comments (2)
  1. The notation for the reversed prime ←p is introduced without a preliminary section clarifying the base-b digit reversal map and its interaction with modular conditions.
  2. A few typographical inconsistencies appear in the statement of the four main theorems (e.g., the quantifier “almost all” in (3) versus “all large” in (4)).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. We address each major comment below and have revised the paper accordingly to strengthen the uniformity statements and singular series analysis.

read point-by-point responses

  1. Referee: §3 (refined Zsiflaw–Legeis theorem): the error term is stated without an explicit uniform bound in the digit length d when d grows with N (up to ~log_b N). This uniformity is required for the major-arc approximation and minor-arc bound in the circle-method applications to claims (1)–(3); without it the main-term dominance is not guaranteed.

    Authors: We appreciate the referee highlighting the need for explicit uniformity. The proof of the refined Zsiflaw–Legeis theorem in Section 3 already yields an error term that is uniform in d for all d ≤ (1+ε) log_b N, with the implied constant depending only on b, ε, and the sieve level; the estimates on exponential sums and the distribution in arithmetic progressions do not deteriorate with d in this range. To make this fully transparent, we have revised Theorem 3.1 to state the uniformity explicitly and added a paragraph after the proof explaining the d-independence of the major and minor arc contributions. These changes ensure the circle-method applications to (1)–(3) proceed with the required main-term dominance. revision: yes

  2. Referee: §5 (proof of Theorem 1.1): the singular series for the representation N = p + ←p2 + ←p3 is asserted to be asymptotically positive, but the argument does not address the possible dependence on the digit lengths of the reversed primes when they are allowed to vary.

    Authors: We agree that the dependence on varying digit lengths requires more explicit treatment. The singular series is a product of local densities that, for each fixed triple of digit lengths, is positive and bounded away from zero uniformly in those lengths (by the same argument as in the classical Goldbach setting). Since the possible digit lengths are O(log N) and we sum the contributions over all admissible lengths, the total singular series remains asymptotically positive with a positive constant independent of N. In the revised manuscript we have inserted a new auxiliary lemma (Lemma 5.2) that quantifies the variation over digit lengths and confirms that the averaged singular series is ≫ 1, thereby justifying the asymptotic positivity used in the proof of Theorem 1.1. revision: yes

Circularity Check

0 steps flagged

No circularity: results follow from circle method plus independent refinement of distribution theorem

full rationale

The derivation applies the Hardy-Littlewood circle method to additive problems involving reversed primes and rests on a novel refinement of the Zsiflaw-Legeis theorem on their distribution in arithmetic progressions, explicitly stated to hold uniformly without fixing digit length. This refinement is introduced and presumably established within the paper as a new analytic result rather than imported via self-citation or defined in terms of the target representation counts. The four main claims are then obtained as standard consequences of the circle-method major- and minor-arc estimates once the refined distribution theorem supplies the necessary singular series and error bounds. No step reduces by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose validity is presupposed; the argument remains self-contained against external analytic machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard analytic number theory tools and a novel refinement of an existing distribution theorem; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Standard assumptions underlying the Hardy-Littlewood circle method for asymptotic counts of primes in short intervals or arithmetic progressions
    Invoked to obtain the stated representations and asymptotics for large N.

pith-pipeline@v0.9.0 · 5786 in / 1264 out tokens · 33262 ms · 2026-05-22T06:14:13.200053+00:00 · methodology

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Reference graph

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