pith. sign in

arxiv: 2606.30188 · v1 · pith:6AEZI3KAnew · submitted 2026-06-29 · 🧮 math.NT

On integers of the form \(p+F_(2^k)+F_q\)

Yang Gao This is my paper

Pith reviewed 2026-06-30 05:16 UTC · model grok-4.3

classification 🧮 math.NT
keywords Fibonacci numbersprimesasymptotic densityRomanoff theoremadditive number theorypowers of two
0
0 comments X

The pith

The set of integers of the form p + F_{2^k} + F_q has positive lower asymptotic density, as does its complement.

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

The paper establishes that positive integers of the form prime plus a Fibonacci number with power-of-two index plus another prime possess positive lower asymptotic density. It further establishes that the integers not of this form also possess positive lower asymptotic density. This extends the classical result of Romanoff on primes plus powers of two by incorporating Fibonacci numbers at specific indices. A sympathetic reader cares because it indicates that the selected Fibonacci numbers do not disrupt the density properties when added to prime sums.

Core claim

The set of positive integers of the form p + F_{2^k} + F_q, where p and q are primes and k ≥ 0, has positive lower asymptotic density. The same holds for the set of integers not of this form.

What carries the argument

Romanoff-type density arguments adapted to the Fibonacci sequence with indices restricted to powers of two.

If this is right

  • The representable set has positive lower asymptotic density.
  • The non-representable set has positive lower asymptotic density.
  • Both sets are therefore infinite.
  • The Fibonacci terms at the chosen indices do not force zero density in either set.

Where Pith is reading between the lines

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

  • Similar results may hold if the power-of-two restriction is relaxed to other sparse sets of indices.
  • The approach could apply to other binary recurrences like Lucas sequences.
  • This suggests the possibility of density results for sums involving multiple Fibonacci numbers.

Load-bearing premise

The prime distribution and Fibonacci properties allow the density arguments to succeed without the specific indices introducing zero-density sets of exceptions.

What would settle it

The discovery of a long arithmetic progression consisting entirely of integers that cannot be expressed as p + F_{2^k} + F_q would falsify the positive density for the representable set.

read the original abstract

In 1934, Romanoff proved that the set of positive integers representable as the sum of a prime and a power of two has positive lower density. Erd\H{o}s later constructed an infinite arithmetic progression of odd integers none of which admits such a representation. Let \(F_n\) be the Fibonacci sequence. In this paper, we prove that the set of integers of the form \(p+F_{2^k}+F_q\), where \(p,q\) are primes and \(k\ge0\), has positive lower asymptotic density. The same holds for the set of integers not of this form.

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 / 1 minor

Summary. The paper proves that the set of positive integers representable in the form p + F_{2^k} + F_q (p, q prime, k ≥ 0) has positive lower asymptotic density, and that its complement likewise has positive lower asymptotic density, by adapting Romanoff-type sieve arguments for the representable set and an Erdős-style arithmetic-progression construction for the complement.

Significance. If correct, the result extends the classical Romanoff–Erdős dichotomy to a sumset involving two restricted subsequences of the Fibonacci numbers. It would show that the indices 2^k and prime q are flexible enough, despite the gcd(F_m, F_n) = F_{gcd(m,n)} relation and Pisano periodicity, to produce a set A whose translates by primes cover a positive-density subset while still leaving a positive-density uncovered set.

major comments (1)
  1. [Main theorem and § on the sieve estimate] The central claim rests on showing that A = {F_{2^k} + F_q} is sufficiently dense in residue classes modulo the primorial to permit a positive lower bound via the Romanoff sieve; the manuscript must explicitly rule out the possibility that the restricted indices force A into a proper subset of residues modulo small m (via the Pisano period or gcd properties), as this would collapse the lower density to zero. No such verification is visible in the abstract or the sketched argument.
minor comments (1)
  1. Notation for the Fibonacci sequence and the range of k should be stated once at the beginning rather than repeated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for an explicit verification in the sieve argument. The concern is well-taken and will be addressed by adding the required check to the manuscript.

read point-by-point responses

  1. Referee: [Main theorem and § on the sieve estimate] The central claim rests on showing that A = {F_{2^k} + F_q} is sufficiently dense in residue classes modulo the primorial to permit a positive lower bound via the Romanoff sieve; the manuscript must explicitly rule out the possibility that the restricted indices force A into a proper subset of residues modulo small m (via the Pisano period or gcd properties), as this would collapse the lower density to zero. No such verification is visible in the abstract or the sketched argument.

    Authors: We agree that the manuscript must contain an explicit verification that the restricted indices do not confine A to a proper subset of residue classes modulo small primorials. In the revised version we will insert a new lemma (placed immediately before the application of the Romanoff sieve) that uses the Pisano period of the Fibonacci sequence modulo m together with the fact that the set of primes q is positive-density in the arithmetic progressions compatible with the period. This lemma will show that F_{2^k} + F_q occupies a positive proportion of the residue classes modulo the primorial that are coprime to the small primes appearing in the sieve, thereby ensuring the lower-density bound remains positive. The argument will be self-contained and will not rely on the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity; direct adaptation of Romanoff/Erdős density arguments

full rationale

The paper states a direct proof that the indicated set has positive lower asymptotic density (and likewise its complement) by adapting known Romanoff-type sieve estimates and Erdős arithmetic-progression constructions to the specific sum p + F_{2^k} + F_q. No load-bearing step reduces the claimed density to a fitted parameter, a self-definition, or a self-citation chain; the argument relies on external facts (Pisano periodicity, gcd(F_m,F_n)=F_gcd(m,n), prime-distribution results) that are not constructed from the target density statement itself. The abstract and described derivation chain therefore remain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available so ledger is minimal; relies on standard facts about primes, Fibonacci recurrence, and asymptotic density.

axioms (1)
  • standard math Standard properties of the Fibonacci sequence and the distribution of primes in arithmetic progressions
    Invoked to adapt density arguments from the power-of-two case

pith-pipeline@v0.9.1-grok · 5617 in / 1032 out tokens · 29869 ms · 2026-06-30T05:16:00.914378+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

30 extracted references · 2 canonical work pages

  1. [1]

    Chen and X.-G

    Y.-G. Chen and X.-G. Sun,On Romanoff’s constant, J. Number Theory106(2004), no. 2, 275–284

    2004

  2. [2]

    Chen and R.-J

    Y.-G. Chen and R.-J. Wang,The sum of a prime and a term of exponential sequences, Acta Math. Sinica (Chinese Ser.)67(2024), no. 2, 259–272

    2024

  3. [3]

    Chen and J.-Z

    Y.-G. Chen and J.-Z. Xu,On integers of the formp+ 2 kr1 1 +· · ·+ 2 krt t , J. Number Theory 258(2024), 66–93. 15

    2024

  4. [4]

    Xu and Y.-G

    J.-Z. Xu and Y.-G. Chen,On integers of the formp+ 2 a2 + 2b2 , Int. J. Number Theory21 (2025), no. 1, 153–175

    2025

  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

    1971

  6. [6]

    Dai and H

    L. Dai and H. Pan,Polynomial analogue of Erd˝ os’ extension of the Romanoff theorem, J. Number Theory279(2026), 45–55

    2026

  7. [7]

    de Polignac,Recherches nouvelles sur les nombres premiers, C

    A. de Polignac,Recherches nouvelles sur les nombres premiers, C. R. Acad. Sci. Paris29 (1849), 738–739

  8. [8]

    Ding,On a problem of Romanoff type, Acta Arith.205(2022), no

    Y. Ding,On a problem of Romanoff type, Acta Arith.205(2022), no. 1, 53–62

    2022

  9. [9]

    Dubickas,Sums of primes and quadratic linear recurrence sequences, Acta Math

    A. Dubickas,Sums of primes and quadratic linear recurrence sequences, Acta Math. Sin. (Engl. Ser.)29(2013), no. 12, 2251–2260

    2013

  10. [10]

    Elsholtz and J.-C

    C. Elsholtz and J.-C. Schlage-Puchta,On Romanov’s constant, Math. Z.288(2018), no. 3-4, 713–724

    2018

  11. [11]

    Elsholtz, F

    C. Elsholtz, F. Luca and S. Planitzer,Romanov type problems, Monatsh. Math.187(2018), no. 2, 267–289

    2018

  12. [12]

    Erd˝ os,On integers of the form2 k +pand some related problems, Summa Brasil

    P. Erd˝ os,On integers of the form2 k +pand some related problems, Summa Brasil. Math.2 (1950), 113–123

    1950

  13. [13]

    G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, Oxford, 1979

    1979

  14. [14]

    K. S. E. Lee,On the sum of a prime and a Fibonacci number, Int. J. Number Theory6 (2010), no. 7, 1669–1676

    2010

  15. [15]

    Liu and M

    Z. Liu and M. Xue,The sum of a prime and a Fibonacci number, Int. J. Number Theory17 (2021), no. 8, 1815–1823

    2021

  16. [16]

    Liu and G

    Z. Liu and G. L¨ u,Density of two squares of primes and powers of2, Int. J. Number Theory 7(2011), no. 5, 1317–1329

    2011

  17. [17]

    M. B. Nathanson,Additive Number Theory: The Classical Bases, Graduate Texts in Mathematics, vol. 164, Springer, New York, 1996

    1996

  18. [18]

    Pan,On the integers not of the formp+ 2 a + 2b, Acta Arith.148(2011), 55–61

    H. Pan,On the integers not of the formp+ 2 a + 2b, Acta Arith.148(2011), 55–61

    2011

  19. [19]

    N. P. Romanoff, ¨Uber einige S¨ atze der additiven Zahlentheorie, Math. Ann.109(1934), 668–678

    1934

  20. [20]

    Sierpi´ nski,Sur un probl` eme concernant les nombresk·2 n + 1, Elem

    W. Sierpi´ nski,Sur un probl` eme concernant les nombresk·2 n + 1, Elem. Math.15(1960), 73–74

    1960

  21. [21]

    ˇSiurys,On integers not of the formF n ±p a, Int

    J. ˇSiurys,On integers not of the formF n ±p a, Int. J. Number Theory12(2016), no. 2, 505–512

    2016

  22. [22]

    Somer,Distribution of residues of certain second-order linear recurrences modulop, in Applications of Fibonacci Numbers, Vol

    L. Somer,Distribution of residues of certain second-order linear recurrences modulop, in Applications of Fibonacci Numbers, Vol. 3, Kluwer Acad. Publ., 1990, pp. 311–324

    1990

  23. [23]

    Sun,On the density of integers of the form2 k +pin arithmetic progressions, Acta Math

    X.-G. Sun,On the density of integers of the form2 k +pin arithmetic progressions, Acta Math. Sin. (Engl. Ser.)26(2010), no. 1, 155–160

    2010

  24. [24]

    Sun and J.-H

    X.-G. Sun and J.-H. Fang,On the density of integers of the form(p−1)2 −n in arithmetic progressions, Bull. Aust. Math. Soc.78(2008), no. 3, 431–436. 16

    2008

  25. [25]

    Sun and M.-H

    Z.-W. Sun and M.-H. Li,On integers not of the formc(2 a + 2b) +p α, Acta Arith.99(2001), 183–201

    2001

  26. [26]

    Wang,On the sum of a Lucas number and a prime, Period

    R.-J. Wang,On the sum of a Lucas number and a prime, Period. Math. Hungar.90(2025), no. 2, 434–439

    2025

  27. [27]

    Wang and Y.-G

    R.-J. Wang and Y.-G. Chen,On the sum of a Fibonacci number and a prime, Int. J. Number Theory19(2023), no. 4, 873–889

    2023

  28. [28]

    Wang,On arithmetic progressions of positive integers avoidingp+F m andq+L n, arXiv:2506.12047 (2025)

    R.-J. Wang,On arithmetic progressions of positive integers avoidingp+F m andq+L n, arXiv:2506.12047 (2025)

    work page arXiv 2025

  29. [29]

    Xu and Y.-G

    J.-Z. Xu and Y.-G. Chen,On the sum of a prime and two Fibonacci numbers, arXiv:2506.03631 (2025)

    work page arXiv 2025

  30. [30]

    Yuan,On integers not of the formc(2 a + 2b) +p α, Acta Math

    P.-Z. Yuan,On integers not of the formc(2 a + 2b) +p α, Acta Math. Sin. (Chin. Ser.)47 (2004), 139–144. 17

    2004