On arithmetic progressions of positive integers avoiding $p+F_m$ and $q+L_n$

Rui-Jing Wang

REVIEW 2 major objections 4 minor 22 references

On arithmetic progressions of positive integers avoiding $p+F_m$ and $q+L_n$

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A single arithmetic progression of positive integers sidesteps both prime-plus-Fibonacci and prime-plus-Lucas representations.

desk verdict A genuine simultaneous Romanoff-type result, explicit and likely correct; the only real gap is an unverified covering-system assertion that a referee can close in an afternoon. read the letter →

arxiv 2506.12047 v1 pith:IGKTR3PG submitted 2025-05-26 math.GM

Rui-Jing Wang This is my paper · ORCID

classification math.GM MSC 11P3211B39

keywords FibonaccinumbersLucasarithmeticprogressionscoveringsystemsprimesRomanov-typeproblems

topics Goldbach's Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that one arithmetic progression of positive integers can avoid both additive forms $p+F_m$ and $q+L_n$, where $p,q$ range over primes and $F_m,L_n$ are Fibonacci and Lucas numbers. The paper displays two integers $M$ and $N$ and proves that every number $Mk+N$ with $k\ge 0$ is expressible neither as a prime plus a Fibonacci number nor as a prime plus a Lucas number. Earlier work gave such arithmetic progressions separately for the Fibonacci case and for the Lucas case, and the paper notes that the combined statement cannot be obtained by intersecting those older progressions. The proof uses covering systems and the Chinese remainder theorem, then rules out all remaining cases through congruences modulo $17$, $19$, and $47$.

What carries the argument

The machinery is a pair of covering systems combined through the Chinese remainder theorem. A covering system is a finite list of congruences $n\equiv a_i\pmod{m_i}$ such that every integer satisfies at least one of them. Lemma 2.1 supplies one system of 33 congruences, with least common multiple 1152, governing Fibonacci indices, and one system of 15 congruences, with least common multiple 576, governing Lucas indices. Since Fibonacci numbers modulo a prime $p_i$ are periodic with period $\chi_f(p_i)=m_i$, and similarly for Lucas numbers, any representation forces the prime summand to be one of finitely many listed primes. The choice of moduli is balanced so that the largest modulus 1152 is just large enough; the paper notes that it uses all primes whose Fibonacci or Lucas period divides 1152, which is what makes the constants large. The final contradictions come from tables of $F_m$ modulo $17,19,47$ and $L_n$ modulo $19,47$.

What would settle it

Write a short program that loops over all residues modulo 1152 and checks whether any residue is missed by the 33 congruences in the first system of Lemma 2.1, and likewise loops over residues modulo 576 for the 15-congruence system; a single uncovered residue would invalidate the construction. Independently, verifying each displayed congruence $F_{m_i k+a_i}\equiv r_i\pmod{p_i}$ and $L_{n_j k+b_j}\equiv s_j\pmod{q_j}$ would confirm the period tables on which the final contradictions rest.

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Extended reading notes

Core claim

The central discovery is that both representation problems can be disposed of by a single congruence construction. Two covering systems, one of 33 congruences for Fibonacci indices and one of 15 congruences for Lucas indices, force the index of any hypothetical representation into a residue class modulo a listed prime. Because the target number and the forced Fibonacci or Lucas term are congruent modulo that same prime, the prime summand must equal the listed prime itself. The argument then reduces to checking finitely many possibilities of the form $x=p_i+F_{m_i k+a_i}$ or $x=q_j+L_{n_j k+b_j}$, and all of these contradict period tables of the Fibonacci and Lucas sequences modulo $17$, $19$, or $47$. Hence the displayed progression $Mk+N$ lies in the set $B$ of integers with neither representation.

Load-bearing premise

The load-bearing premise is Lemma 2.1's assertion that the two displayed lists of congruences cover all integers modulo 1152 and 576, a check the proof leaves to a computer calculation; if either list misses one residue class, the reduction of every Fibonacci or Lucas index to finitely many primes collapses.

Editorial extensions

If this is right

The set of positive integers having neither a prime-plus-Fibonacci nor a prime-plus-Lucas representation contains an infinite arithmetic progression.
There is no structural obstruction to merging the Fibonacci and Lucas avoidance problems into one Chinese-remainder construction.
The proof yields separate progressions for the Fibonacci-only and Lucas-only avoiding sets, with much smaller parameters in the Lucas-only case (Corollaries 4.1 through 4.3).
Because the progression is explicit, membership of any particular term can be checked; the paper reports a computer verification of $Mk+N\in B$ for $0\le k\le 10^5$.
Problem 1.4 asks whether the set of integers representable in both ways has positive lower asymptotic density, extending the same quantitative framework.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same covering-system pattern may generalize to other linear recurrences or to sums of a prime with two independent fast-growing sequences, provided the moduli are chosen from the periods of the sequences modulo primes.
The paper remarks that the large size of $M$ and $N$ comes from using all primes whose Fibonacci or Lucas period divides 1152; a computer search over alternative covering systems could plausibly produce much smaller progressions.
Since the deciding contradictions use only the small moduli 17, 19, and 47, a natural testable extension is to look for covering systems whose forced primes can be separated by even smaller moduli, which would shrink the constants and might improve quantitative density bounds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The main thing to know: this paper does what it claims. It gives explicit M and N such that Mk+N is never a prime plus a Fibonacci number and never a prime plus a Lucas number. That simultaneous version is new, and Remark 1.2 is correct that it is not just the intersection of the existing one-sided arithmetic progressions. The construction is classical Erdős covering-system material, but balancing 33 Fibonacci congruences with 15 Lucas congruences so that the same moduli work for both sequences is real work. The proof after Lemma 2.1 is straightforward, the three F_tables and two L_tables check out, and the contradiction arguments are easy to follow.

The load-bearing step is Lemma 2.1. Both covering systems must cover every residue class modulo 1152 and 576, respectively. The proof only says 'By calculation, one can easily verify.' No code, no certificate, no enumeration is included. If either system misses one residue, the chain from a representation x=p+F_m or x=q+L_n to the finite prime lists at (3.2)–(3.3) breaks. I looked at the two systems by splitting into 2-adic and 3-adic parts; coverage looks plausible. Direct enumeration is trivial at these moduli, so this is fixable, but the manuscript as submitted does not contain that check. An editor should not have to take the author's word on a computationally decisive step.

Minor issue: the huge moduli are used as primes without proof or certified primality. Standard tests handle them, but it is another unstated computational claim.

No circularity: the progression comes from CRT on the chosen congruences, and the contradiction tables do not presuppose the conclusion. The citation pattern is fine; [3] supplies inspiration and reference values, and self-citation is justified. The paper is honest about its trial-and-error design in Remark 3.1.

For whom: researchers in additive number theory working on Romanoff-type problems or covering-system constructions. Not field-shaping, but a solid, explicit contribution. It deserves a serious referee. I would send it out with a request that the author supply an independent verification of Lemma 2.1—a coverage table or a few lines of code. If that lands, the theorem is proved.

I'd probably cite it if my own work touched this exact question.

Referee Report

2 major / 4 minor

Summary. The paper proves Theorem 1.1: for two explicitly displayed very large integers M and N, every term of the arithmetic progression Mk+N (k≥0) lies in the set B, meaning it is simultaneously not of the form p+F_m for any prime p and Fibonacci number F_m, and not of the form q+L_n for any prime q and Lucas number L_n. The proof constructs a system of 48 congruences (33 for the Fibonacci branch, 15 for the Lucas branch), invokes two covering systems modulo 1152 and 576 to reduce the infinite families of indices to finitely many cases, uses the Chinese remainder theorem to identify the solution set as the progression Mk+N, and then rules out each finite case by tables of F_m and L_n modulo 17, 19, and 47. Corollaries give analogous one-sided progressions for B_f and B_l.

Significance. If fully verified, the result is a clean and nontrivial addition to the Romanoń/de Polignac literature: it gives an explicit arithmetic progression whose terms avoid both p+F_m and q+L_n simultaneously, going beyond the earlier one-sided results of Jones, Ismailescu, Šiurys, and Ismailescu–Shim. The overall proof structure is transparent: once the covering systems and the modular identities are granted, the final contradiction is a finite check using hand-verifiable tables. The paper also states explicit corollaries for the one-sided sets. Its main weakness is that the load-bearing computational material—the covering systems and the many modular identities involving very large primes—is asserted without code, certificate, or reproducible verification, so the result is not machine-checkable as written.

major comments (2)

[Lemma 2.1] The two covering systems are load-bearing, since they are what restrict the infinitely many Fibonacci and Lucas indices to the finite lists used in (3.2) and (3.3), but the proof is only the sentence 'By calculation, one can easily verify Lemma 2.1.' No certificate is provided for either system. If one residue class modulo 1152 (resp. 576) were missed, the argument would collapse. Please provide an explicit certificate—for example, for each residue class modulo 1152 and 576, list which congruence covers it, or include a short reproducible enumeration script.
[Section 3, displayed congruences and equation (3.1)] The reduction to p=p_{i_0} and q=q_{j_0} in (3.2) and (3.3) relies on the displayed identities F_{m_i k+a_i} ≡ r_i (mod p_i) for 1≤i≤33, L_{n_j k+b_j} ≡ s_j (mod q_j) for 1≤j≤15, and the equalities m_i=χ_f(p_i), n_j=χ_l(q_j). These are asserted with 'some help of a computer' but no derivation, code, or certificate is included. The argument also requires all the p_i and q_j to be prime, and this is not verified for the very large moduli such as 115561578124838522881 and 1974737795746080149567. Please supply a reproducible verification, ideally a short script or explicit certificates for each congruence and each primality assertion.

minor comments (4)

[Section 3, equation (3.1)] The Chinese remainder theorem step silently assumes that the combined system of 48 congruences is consistent. The repeated moduli in the two lists (2, 3, 7, 23, 47, 19, 17, 2207) do have matching residues, but this should be stated explicitly.
[Proof of Theorem 1.1, case analysis] In a few cases the phrase 'a contradiction to Table X' is compressed: the target residue does appear elsewhere in the table, and the contradiction only follows after restricting to the possible residue classes of the index. This occurs, for example, in the m_i=16 case (2 mod 17 appears in Table 1) and in the m_i=36 and m_i=1152, p_i=1974737795746080149567 cases. Please spell out the residue-class restriction in each such case.
[Corollaries 4.1–4.3] The corollaries are stated with explicit M and N but no proof is given beyond 'can be deduced.' Since they use different covering systems and different congruences, they should either be proved explicitly or clearly labeled as computational claims requiring the same verification as the main theorem.
[Throughout] There are minor typographical issues: 'inevitablely' in Remark 1.3, 'balence' in Section 3, and inconsistent spacing in the displayed M and N. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the progression Mk+N is derived from explicit covering systems, CRT, and finite contradiction tables, not from the conclusion.

full rationale

This paper's derivation is self-contained rather than circular. Theorem 1.1 is obtained from explicitly displayed covering systems in Lemma 2.1, explicit Fibonacci and Lucas congruence properties, the Chinese remainder theorem to define the progression Mk+N via (3.1), and finite contradiction tables modulo 17, 19, and 47. No fitted parameter is renamed as a prediction: M and N are produced by CRT from the congruences, and the theorem is then proved for all k by the case analysis after (3.2) and (3.3), not by checking the displayed progression. The only self-citation, [3] by Chen and Wang including the present author, supplies inspiration and reference values but no theorem on which the proof depends. Lemma 2.1 is asserted by direct calculation and the Fibonacci/Lucas congruences are asserted with computer help, which are computational-verification gaps rather than circular reductions; they do not make the argument equivalent to its inputs. Thus no circular step reduces the derivation to its own conclusion or to a self-citation chain.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The theorem is a constructive existence proof. Its load-bearing choices are the hand-selected covering systems and prime moduli; these are not derived from an external benchmark, but they are fully explicit and checkable, so the burden is verification rather than fitting.

free parameters (2)

Fibonacci-branch covering system entries (a_i, m_i, r_i, p_i), i=1..33 = Explicit list in Lemma 2.1 and the proof
Hand-selected after multiple attempts, as described in Remark 3.1; the proof's contradiction cases depend on these exact residue choices.
Lucas-branch covering system entries (b_j, n_j, s_j, q_j), j=1..15 = Explicit list in Lemma 2.1 and the proof
Hand-selected by trial and error; Remark 3.1 explains that a 14-entry system was discarded because it finished the Lucas step too early and broke the Fibonacci step.

assumptions (5)

ad hoc to paper The two residue systems in Lemma 2.1 are covering systems.
Stated in Lemma 2.1 with "By calculation, one can easily verify"; no derivation or code is provided. If any residue class is missing, the reduction to finitely many primes fails.
domain assumption Every listed p_i and q_j is prime.
Used to conclude p equals p_i from p congruent to 0 modulo p_i in equations (3.2) and (3.3). No primality certificates or factorizations are given.
domain assumption The periodic congruences for F_m and L_n listed before the proof hold for all k.
These are finite assertions modulo primes with periods dividing 1152; they are asserted without demonstration, though readily checkable.
standard math Tables 1 through 5 give the correct residues of F_m modulo 17, 19, 47 and L_n modulo 19, 47.
Finite computations underlying the contradiction step; not derived in the text but easily reproducible.
standard math The Chinese remainder theorem applies to the simultaneous congruences in (3.1).
The moduli are primes and the repeated primes have compatible residues; CRT gives the explicit M and N.

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Pith. "Pith review of On arithmetic progressions of positive integers avoiding $p+F_m$ and $q+L_n$." pith.science (2026). https://pith.science/paper/IGKTR3PG

@misc{pith2026250612047,
  author       = {Pith},
  title        = {Pith review of: On arithmetic progressions of positive integers avoiding $p+F_m$ and $q+L_n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGKTR3PG}},
  note         = {Machine review of arXiv:2506.12047}
}

abstract

In this paper, it is proved that there is an arithmetic progression of positive integers such that each of which is expressible neither as $p+F_m$ nor as $q+L_n$, where $ p,q $ are primes, $ F_m $ denotes the $ m $-th Fibonacci number and $ L_n $ denotes the $ n $-th Lucas number.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [3]

Chen, R.-J

Y.-G. Chen, R.-J. Wang, The sum of a prime and a term of exponent ial sequences (Chinese), Acta Math. Sin. 67 (2024) 259–272

work page 2024

[2] [1]

Y.-G. Chen, Y. Ding, On a conjecture of Erd˝ os, C. R. Math. Acad. Sci. Paris 360 (2022) 971–974

work page 2022

[3] [2]

Y.-G. Chen, Y. Ding, Quantitative results of the Romanov type re presentation function, Q. J. Math. 74 (2023) 1331–1359

work page 2023

[4] [4]

Chen, J.-Z

Y.-G. Chen, J.-Z. Xu, On integers of the form p + 2kr1 1 + · · · + 2krt t , J. Number Theory 258 (2024) 66–93

work page 2024

[5] [5]

Crocker, On the sum of a prime and two powers of two, Paciﬁc J

R. Crocker, On the sum of a prime and two powers of two, Paciﬁc J. Math. 36 (1971) 103–107

work page 1971

[6] [6]

de Polignac, Six propositions arithmologiques d´ eduites du crible d’Eratosth´ ene, Nouv

A. de Polignac, Six propositions arithmologiques d´ eduites du crible d’Eratosth´ ene, Nouv. Ann. Math. 8 (1849) 423–429

work page

[7] [7]

de Polignac, Recherches nouvelles sur ies nombres primiers, C

A. de Polignac, Recherches nouvelles sur ies nombres primiers, C. R. Acad. Sci. Paris 29 (1849), 738–739

work page

[8] [8]

Elsholtz and J.-C

C. Elsholtz and J.-C. Schlage-Puchta, On Romanov’s constant, Math. Z. 288 (2018) 713–724

work page 2018