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

Yang Gao

REVIEW 1 major objections 1 minor 30 references

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

T0 review · 1 major / 1 minor · reviewed 2026-06-30 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The set of integers of the form p + F_{2^k} + F_q has positive lower asymptotic density, as does its complement.

desk verdict Gao gives a Fibonacci-index variant of Romanoff's density result, but the restricted indices raise real questions about whether the sums stay dense enough modulo small primes. read the letter →

arxiv 2606.30188 v1 pith:6AEZI3KA submitted 2026-06-29 math.NT

Yang Gao This is my paper · ORCID

classification math.NT

keywords FibonaccinumbersprimesasymptoticdensityRomanofftheoremadditivenumbertheorypowersoftwo

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

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.

What carries the argument

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

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.

Watch

Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main claim is that the integers of the form p + F_{2^k} + F_q have positive lower asymptotic density, and so do the integers outside that set. This is a direct swap of the power-of-two term in Romanoff's 1934 theorem for a sum of two Fibonacci numbers, one at a power-of-two index and one at a prime index.

The new element is the choice of those specific indices rather than arbitrary or power-of-two indices alone. The paper sets up the problem cleanly by recalling the classic positive-density result and the Erdős arithmetic-progression complement, then states the analogous statement for the Fibonacci case.

The technique appears to be an adaptation of the usual sieve or density estimates used for prime-plus-power-of-two sums. That part is straightforward in outline.

The soft spot is the distribution of A = {F_{2^k} + F_q}. Fibonacci numbers obey gcd(F_m, F_n) = F_gcd(m,n) and are periodic modulo any fixed m. Restricting the indices to powers of two and to primes can force the possible residues of those sums into a proper subset of the residues modulo small primes or primorials. If A misses too many classes, the lower-density sieve bound can fail or the construction of a positive-density complement can break. The abstract gives no indication that this modular obstruction was checked or bypassed.

If the full argument contains explicit estimates showing that A still hits enough residue classes, the result is fine. Otherwise the claim rests on an unverified adaptation.

This is for people who work on additive bases with restricted sequences such as Fibonacci numbers. A reader already familiar with Romanoff-type arguments would see the value in the index choice and could check the details.

It deserves a serious referee to verify the sieve estimates and the handling of the modular constraints.

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)

[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)

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

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

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

assumptions (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

how reviews work

0 comments

Cite this review

Pith. "Pith review of On integers of the form \(p+F_{2^k}+F_q\)." pith.science (2026). https://pith.science/paper/6AEZI3KA

@misc{pith2026260630188,
  author       = {Pith},
  title        = {Pith review of: On integers of the form \(p+F_2^k+F_q\)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6AEZI3KA}},
  note         = {Machine review of arXiv:2606.30188}
}

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.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 2 canonical work pages

[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]

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]

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]

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]

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]

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]

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]

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

Show all 30 references

[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]

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]

Elsholtz, F

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

2018
[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]

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

1979
[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]

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]

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]

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

1996
[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]

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

1934
[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]

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]

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]

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]

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]

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]

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]

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]

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)

2025
[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)

2025
[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

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.

[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