Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

Daniel Duverney; Yohei Tachiya; Yuta Suzuki

REVIEW 1 major objections 3 minor 13 references

Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

T0 review · 1 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For any algebraic integer with all other conjugates inside the unit disk, five associated power-series values are linearly independent over the field they generate.

desk verdict New linear independence theorem with a clever Erdős-type construction, but the proof has a genuine gap at the end: the assertion |D_k| → ∞ is false, so the contradiction isn't established. read the letter →

arxiv 1908.07290 v1 pith:2TIYMRQ3 submitted 2019-08-20 math.NT

Daniel Duverney , Yuta Suzuki , Yohei Tachiya This is my paper · ORCID

classification math.NT MSC 11J7211A41

keywords linearindependenceLambertseriesFibonaccinumbersLucasPisotreciprocalsumsChineseremaindertheorem

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 a linear independence theorem for special power series whose coefficients count divisors coming from a fixed sequence of pairwise coprime odd integers. When that sequence is the squares of the primes, these series become sums of reciprocals of Fibonacci and Lucas numbers with prime-square indices. As a concrete consequence, the three numbers $1$, $\sum_p 1/F_{p^2}$, and $\sum_p 1/L_{p^2}$ are linearly independent over $\mathbb{Q}(\sqrt{5})$. The result refines a classical irrationality method into a general linear-independence statement over number fields.

What carries the argument

The key machinery is a block-construction method. For each large parameter $k$, the proof builds a system of simultaneous congruences modulo the first many terms of $\{n_\ell\}$, chosen so that the Chinese remainder theorem produces an arithmetic progression $G(k)$ of length $B_k$ such that, for each position $m$ in a block of length $8k-4$, the coefficients $b_j(\gamma+m)$ take prescribed powers of 2 (or zero), while outside the block they are bounded by $\xi^{|i|}$ for any fixed $\xi>1$. Inserting these controlled values into $f_j(\alpha^{-1})$ isolates a dominant term proportional to $(2/\alpha^8)^k$; a norm argument over $\mathbb{Q}(\alpha)$ then forces a contradiction if the five numbers were linearly dependent. The construction uses inclusion-exclusion and the prime number theorem to show that most elements of $G(k)$ avoid divisibility by the large terms of the sequence.

What would settle it

Let $\alpha=(1+\sqrt{5})/2$ and take $n_\ell = p_\ell^2$ (squares of odd primes). Compute $S_1=\sum_p 1/F_{p^2}$ and $S_2=\sum_p 1/L_{p^2}$ to high precision and use lattice reduction over the ring of integers of $\mathbb{Q}(\sqrt{5})$ to search for a nontrivial relation $a + b S_1 + c S_2 = 0$ with small coefficients. The theorem predicts no such relation exists; a discovered relation would disprove the corollary.

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Extended reading notes

Core claim

The central discovery is Theorem 1.1: if $\alpha$ is an algebraic integer with $|\alpha|>1$ whose conjugates over $\mathbb{Q}$ other than itself and its complex conjugate lie in the open unit disk, then the five numbers $1, f_1(\alpha^{-1}), f_2(\alpha^{-1}), f_3(\alpha^{-1}), f_4(\alpha^{-1})$ are linearly independent over $\mathbb{Q}(\alpha)$. The functions $f_j(z)=\sum_{n\geq 1} b_j(n)z^n$ are defined through integer coefficients $b_j(n)$ that count, among the divisors of $n$ belonging to a fixed sequence $\{n_\ell\}$ of odd pairwise coprime integers, those in prescribed residue classes modulo 4. In the special case $\alpha=(1+\sqrt{5})/2$ and $\{n_\ell\}$ the squares of odd primes, the combinations $f_1\pm f_3$ (with appropriate normalizations) reduce to the sums $\sum_p 1/F_{p^2}$ and $\sum_p 1/L_{p^2}$, giving the abstract's example. The proof also yields Corollaries 1.1 and 1.2, which generalize the earlier irrationality theorem to linear independence of four Lambert-type series over $\mathbb{Q}$ and of three reciprocal Lucas-sequence sums over $\mathbb{Q}(\alpha)$.

Load-bearing premise

The sequence of integers $\{n_\ell\}$ must be pairwise coprime (condition (H1)); without this, the Chinese remainder theorem step and the inclusion-exclusion count of 'bad' positions in the block construction both fail.

Editorial extensions

If this is right

For every integer $m \geq 2$, the three numbers $1$, $\sum_p 1/F_{p^m}$, and $\sum_p 1/L_{p^m}$ are linearly independent over $\mathbb{Q}(\sqrt{5})$.
For any integer $t$ with $|t|>1$, the four numbers $1$, $\sum 1/(t^{n_\ell}-1)$, $\sum 1/(t^{n_\ell}+1)$, $\sum t^{n_\ell}/(t^{2n_\ell}-1)$ are linearly independent over $\mathbb{Q}$ for every sequence satisfying (H1) and (H2).
The theorem applies to all Pisot numbers and complex Pisot numbers, giving linear independence over their respective fields.
The proof provides a new general method for proving linear independence of values of Lambert-type series, beyond the particular Fibonacci/Lucas setting.

Reading between the lines

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

The block-construction argument is likely to extend to sequences defined by any nondegenerate Lucas sequence of the first and second kind, since only the Binet form is used in relating the Lambert series to $f_j$.
One could replace the four residue classes modulo 4 by more classes, obtaining linear independence of more than five numbers at the cost of longer blocks and sharper estimates.
The norm argument's rate of decay suggests a quantitative lower bound for linear forms in these numbers (an independence measure) is obtainable by the same method.
A natural testable extension is to algebraic integers whose conjugates lie in a prescribed region beyond the unit disk; the method's estimates would need adjusting.

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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

Here's my read. Theorem 1.1 is a real extension of Erdős's irrationality method to a simultaneous linear independence statement for four Lambert-type series over Q(α). The block construction in Section 2 is intricate and, as far as I can tell, sound; the Fibonacci/Lucas applications are natural and the example with p^2 is attractive. No circularity, no fitted constants.

The soft spot is at the end of the proof of Theorem 1.1. After (3.1), the authors define D_k and claim that |D_k| → ∞ since the ρ_j are not all zero. That claim is false as written. D_k = kA + B, with A = Σ j ρ_j α^{-j} (up to a harmless factor) and B a similar constant. If A = 0 but B ≠ 0, then D_k is a nonzero constant, and |D_k| doesn't go to infinity. That case is actually harmless: |Θ_k| is then about |B|(2/|α|^8)^k, still nonzero, and the bound |Θ_k| < k^5(2/|α|^8)^k holds. The real problem is A = B = 0, so D_k ≡ 0. Then Θ_k = O((2/|α|^8)^k) and the proof gives no lower bound; Θ_k could vanish for large k. The norm argument needs |Θ_k| > 0 to get a nonzero rational integer N_k. The authors give no reason why A and B can't both vanish for a nontrivial relation; in a field of degree > 1, the equations Σ j ρ_j α^{-j} = 0 and the corresponding constant equation have nonzero solutions. So the contradiction is not established.

The reader's report missed this, giving MODERATE confidence and ACCEPT. I think that's too optimistic. The result may well be true, and the gap might be fixable—for example, by showing that if the first few leading terms vanish, a different choice of the ρ_j or a limiting argument still gives a nonzero Θ_k. But as written, the proof is incomplete.

Bottom line: this deserves a serious referee, because the construction is substantial and the theorem is likely correct. I would not accept the current version; request a revision that either proves D_k cannot be identically zero or handles that case separately. I'd bring it to a reading group to discuss the gap, but I wouldn't cite it in its present form.

Referee Report

1 major / 3 minor

Summary. The paper proves a linear-independence statement: if α is an algebraic integer with |α|>1 and all conjugates except possibly α and its complex conjugate lie in the open unit disk, then 1, f1(α^{-1}), f2(α^{-1}), f3(α^{-1}), f4(α^{-1}) are linearly independent over Q(α), where the fj are the Lambert-series generating functions in (1.2). The proof combines an Erdős-type simultaneous-congruence construction that forces prescribed divisor-count values on long intervals with a Pisot-number norm argument. Applications include linear independence over Q(√5) of 1, Σ_p 1/F_{p^2}, and Σ_p 1/L_{p^2}, and more generally of reciprocal sums over Fibonacci and Lucas numbers indexed by pairwise coprime sequences.

Significance. If the proof is correct, the result is a substantial generalization of Erdős’s irrationality theorem and gives new linear-independence results for Lambert series and reciprocal sums of Fibonacci and Lucas numbers. The block construction in Lemmas 2.1–2.3 is intricate and largely self-contained, and the use of the Pisot condition via the Siegel–Chamfy lower bound is elegant. The paper is not accompanied by machine-checked proofs, but the algebraic computations in Section 3 are explicit and checkable. The main concern is a gap in the nonvanishing argument for Θ_k, which is load-bearing for the norm contradiction.

major comments (1)

[§3, Eq. (3.1)–(3.6)] The assertion after (3.1) that |D_k| → ∞ because the ρj are not all zero is false as written. From (3.1) one has D_k = kA+B with A = α^{-4} Σ_{j=1}^4 jρjα^{4-j} and B = Σ_{j=1}^4 ρj(α^{j-4-2εj} - (α^2+1)(α^4+1)α^{1-εj}/(α^8-2)). If A=0 and B≠0, then D_k is a nonzero constant and does not tend to infinity; if A=B=0, then D_k≡0. The proof gives no reason why a nontrivial relation (ρj) satisfying the assumed dependence cannot lie in this kernel, and the system A=B=0 is a homogeneous linear system of two equations in four unknowns over Q(α) with nonzero solutions. This affects the crucial lower bound 0<|Θ_k| in (3.6): if D_k=0 then Θ_k is only O((2/|α|^8)^k), and the argument that Θ_k does not vanish collapses. Even in the constant case A=0, B≠0, the claimed growth is not available, and no separate argument is supplied to prove Θ_k≠0 for all large k. The norm contradiction in (3.3) requires N_k to be a nonzero integer; without a valid nonvanishing proof for Θ_k, the contradiction is not established.

minor comments (3)

[§3, Proposition 3.1] Proposition 3.1 refers to “E1 and E2” and “one of E1 and E2,” but the sets defined in (1.8) are E1 and E3; this is a typo that should be corrected.
[§2.1, after (2.9)] In the displayed inequality just before (2.9), the expression “k42k−1” should presumably be “k·4·2^{k-1}” or similar; please clarify the notation.
[§2.1, Eq. (2.17)] The inequality B_k ≥ 128k is stated after using (2.16); the derivation would be clearer if the intermediate step A_k ≥ 1 or δ_k < 1 were explicitly noted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is derived self-containedly from the stated hypotheses, with only independent external inputs.

full rationale

The paper's derivation chain is self-contained. The main theorem is proved by assuming, for contradiction, a linear relation over Q(alpha) among 1 and the four f_j(alpha^-1) values; the proof then constructs, via Lemmas 2.2-2.5 and Proposition 3.1, a point gamma_0 where the coefficients b_j take prescribed values, derives the asymptotic (3.1), and obtains a contradiction from the norm inequality (3.3). None of these steps uses the target linear independence as an input, and no parameter is fitted to a data subset. The sequence {n_l} is governed by the stated hypotheses (H1) and (H2), and the proof uses the Chinese Remainder Theorem, the Prime Number Theorem, and the independent Pisot-number results of Siegel and Chamfy; these are external mathematical facts, not conclusions derived from the theorem. The self-citations in the paper ([5], [6], [13]) appear only in Remark 1.1 and historical context; they are not load-bearing for Theorem 1.1 or its corollaries. The reviewer-flagged concern about the assertion '|D_k| -> infinity' after (3.1) is a possible correctness gap in the proof, not a circularity: even if that assertion needs further justification, the argument does not reduce to its conclusion or to a fitted input. Accordingly, no circular step is present, and the score is 0.

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

All non-explicit inputs are standard theorems or the paper's own explicit hypotheses. There are no fitted constants and no ad hoc entities. The only non-trivial external facts are the PNT and the Pisot number bounds, both cited to the literature.

assumptions (5)

domain assumption Sequence {n_l} satisfies (H1) pairwise coprime and (H2) ∑ 1/n_l converges.
Explicit hypotheses of the theorem; used throughout, especially the coprimality for the CRT in (2.2)-(2.3) and convergence for defining δ_k and ν_k in Section 2.
domain assumption α is an algebraic integer with |α|>1 and all conjugates except itself and its complex conjugate inside the unit disk.
Hypothesis of Theorem 1.1; used to get |α|^8 > 2 via Siegel/Chamfy and to make the conjugate estimates in Section 3 work.
standard math Prime Number Theorem (π(x) ~ x/log x) and the bound on the number of pairwise coprime integers ≤ x by the number of primes ≤ x.
Used in Lemma 2.2's third step to estimate the count of n_l in F3(k); the PNT is cited and the coprimality bound is stated as 'clearly' true.
standard math Chinese Remainder Theorem.
Used to produce η_k and to count the γ_i satisfying simultaneous congruences in Lemma 2.2.
standard math Siegel's theorem that the smallest Pisot number is θ0 ≈ 1.3247 and Chamfy's complex analogue with minimal modulus sqrt(θ0), yielding |α| ≥ sqrt(θ0).
Cited at the start of Section 2; used to assert |α|^5 > 2, hence |α|^8 > 2, a key inequality in the approximation argument.

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Pith. "Pith review of Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers." pith.science (2026). https://pith.science/paper/2TIYMRQ3

@misc{pith2026190807290,
  author       = {Pith},
  title        = {Pith review of: Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2TIYMRQ3}},
  note         = {Machine review of arXiv:1908.07290}
}

abstract

The aim of this paper is to give linear independence results for the values of certain series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $\{n_\ell\}_{\ell\geq1}$. For example, the three numbers \[ 1,\qquad\sum_{p\text{:prime}}^{}\frac{1}{F_{p^2}},\qquad\sum_{p\text{:prime}}^{}\frac{1}{L_{p^2}} \] are linearly independent over $\mathbb{Q}(\sqrt{5})$, where $\{F_n\}$ and $\{L_n\}$ are the Fibonacci and Lucas numbers, respectively.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Andr´ e-Jeannin, Irrationalit´ e de la somme des inverses de certaines suites r´ ecurrentes

R. Andr´ e-Jeannin, Irrationalit´ e de la somme des inverses de certaines suites r´ ecurrentes. C. R. Acad. Sci. Paris 308(1989), 539–541

work page 1989

[2] [2]

Bundschuh and K

P . Bundschuh and K. V¨ a¨ an¨ anen,Arithmetical investigations of a certain inﬁnite product . Compos. Math. 91(1994), 175–199

work page 1994

[3] [3]

Bundschuh, Arithmetical investigations of particular Wynn power seri es

P . Bundschuh, Arithmetical investigations of particular Wynn power seri es. Hardy-Ramanujan J. 31(2008), 14–27

work page 2008

[4] [4]

Chamfy, F onctions m´ eromorphes dans le cercle-unit´ e et leurs s´ eries de Taylor

C. Chamfy, F onctions m´ eromorphes dans le cercle-unit´ e et leurs s´ eries de Taylor. Ann. Inst. Fourier (Greno- ble) 8(1958), 211–262. https://doi.org/10.5802/aif.78

work page doi:10.5802/aif.78 1958

[5] [5]

Duverney, Irrationalit´ e de la somme des inverses de la suite de Fibonacci

D. Duverney, Irrationalit´ e de la somme des inverses de la suite de Fibonacci. Elem. Math. 52(1997), 31–36. https://doi.org/10.1007/s000170050008

work page doi:10.1007/s000170050008 1997

[6] [6]

Duverney and I

D. Duverney and I. Shiokawa, On series involving Fibonacci and Lucas numbers I . In: Dio- phantine Analysis and Related Fields, American Institute o f Physics, New Y ork, 2008, pp. 62–76. https://doi.org/10.1063/1.2841912

work page doi:10.1063/1.2841912 2008

[7] [7]

Erd˝ os,On the irrationality of certain series

P . Erd˝ os,On the irrationality of certain series . Math. Student 36(1968), 222–226

work page 1968

[8] [8]

Garth, Complex Pisot numbers of small modulus

D. Garth, Complex Pisot numbers of small modulus . C. R. Math. Acad. Sci. Paris 336(2003), 967–970. https://doi.org/10.1016/S1631-073X(03)00236-X

work page doi:10.1016/s1631-073x(03)00236-x 2003