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The paper proves that for a strictly increasing integer sequence a_n with limsup a_n^{1/φ^n} = ∞, the series Σ 1/(a_n a_{n+1}) is irrational, and that the golden-ratio exponent is sharp: for any C>1 a sequence can have lim a_n^{1/φ^n}=C yet

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2026-08-03 06:56 UTC pith:YYW2XEEV

load-bearing objection Real solution to an old Erdős–Graham problem, but the theorem statement overreaches: it says non-decreasing while the proof needs strictly increasing. the 1 major comments →

arxiv 2601.21442 v3 pith:YYW2XEEV submitted 2026-01-29 math.NT math.CA

Irrationality of rapidly converging series: a problem of ErdH{o}s and Graham

Kevin Barreto , Jiwon Kang , Sang-hyun Kim , Vjekoslav Kova\v{c} , Shengtong Zhang This is my paper
classification math.NT math.CA MSC 11J7240A05
keywords Erdős–Graham problemirrationality of seriesgolden ratioAhmes seriesCantor seriesrapid convergenceMahler's criteriongrowth exponent
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Erdős and Graham asked in 1980 how fast a strictly increasing integer sequence must grow to force the sum of reciprocals of consecutive products, Σ 1/(a_n a_{n+1}), to be irrational. The paper answers this by proving that growth with limsup a_n^{1/φ^n} = ∞, where φ is the golden ratio, is sufficient; the original liminf condition with base 2 is therefore more than enough. It also proves the boundary is sharp: for every finite C>1 there exists a strictly increasing integer sequence with lim a_n^{1/φ^n}=C for which the series sums to a rational number. The result extends to weighted products of d consecutive terms, with the golden ratio replaced by the root of a polynomial determined by the weights. A sympathetic reader should take away that the exact exponential growth threshold for this classical irrationality problem is now known.

Core claim

On the paper's own terms, the central discovery is Theorem 2: when d=2, any monotonically increasing integer sequence with limsup a_n^{1/φ^n}=∞ makes Σ 1/(a_n a_{n+1}) irrational, while for every C∈(1,∞) there is a strictly increasing integer sequence with lim a_n^{1/φ^n}=C whose same series is rational. Thus the original Erdős–Graham question has a positive answer, and the exponent φ is optimal at the level of a finite limit. The proof obtains this from a broader weighted theorem (Theorem 3): with weights w_0,...,w_{d-1} and c_w the unique positive root of P_w(x)=(x-1)∑ w_j x^j - W x^{d-1}, the condition limsup a_n^{1/c_w^n}=∞, together with a mild polynomial lower bound on the weighted pro

What carries the argument

The load-bearing mechanism is a classical irrationality criterion of Mahler: if D_N times the first N terms is an integer for all N, then a rational sum would force D_N times the tail to stay bounded away from zero; proving the tail product has a subsequence tending to zero therefore rules out rationality. The paper's novel input is Lemma 12, a 'local peaks' estimate: at indices where µ_{n+1}=log(a_{n+1})/c^{n+1} beats every earlier µ_k by a factor 1+1/n^2, the quotient D_N/x_{N+1} decays super-polynomially. Around these peaks, dyadic-block estimates (Lemma 10) control the tail sums, and Borel's lemma guarantees infinitely many peaks from the limsup hypothesis. The growth exponent c_w is cho

Load-bearing premise

The load-bearing premise is that the sequence terms are positive integers and non-decreasing: the proof forms the integer denominator D_N = ∏_{k=1}^N a_k^W and invokes Mahler's criterion, which would not apply if the terms were arbitrary real numbers.

What would settle it

A counterexample to the main theorem would be a strictly increasing integer sequence with limsup a_n^{1/φ^n}=∞ for which Σ 1/(a_n a_{n+1}) is rational. The paper's own negative constructions all satisfy lim a_n^{1/φ^n}=C<∞, so the limsup case is exactly the untested boundary; one could search computationally for such a sequence using continued-fraction or integer-relation methods on the partial sums.

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If this is right

  • The original Erdős–Graham criterion is confirmed: lim inf a_n^{1/2^n}>1 suffices for irrationality of Σ 1/(a_n a_{n+1}), and the exponent can be lowered to φ when the limsup, rather than liminf, is used.
  • For the d-term products, the critical growth exponent is ψ, the root of ψ^d=ψ^{d-1}+1; this gives a whole family of new irrationality results for rapidly converging reciprocal-product series.
  • The negative half shows that no finite constant C can replace the condition limsup=∞: at every fixed exponential growth rate C the series can be rational, so the boundary between the two regimes is exactly at divergence of a_n^{1/φ^n}.
  • The weighted generalization gives irrationality for series with numerators b_n growing slower than a power, as long as the weighted product denominator grows like n^{1+τ} and the sequence's c_w-th root grows unboundedly.

Where Pith is reading between the lines

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

  • Because the analytic core (Proposition 11) is proved for real sequences and the integrality of a_n enters only at the final application of Mahler's criterion, it is natural to expect that the irrationality conclusion genuinely depends on the terms being integers; constructing real monotone sequences with the same growth and rational sums would confirm this separation.
  • The local-peak strategy is not tied to the specific form of the product denominators; it may transfer to other rapidly convergent series whose denominators are built from several successive sequence values, such as sums of reciprocals of a_n^2+a_{n+1}^2 or a_n+a_{n+1}.
  • For general integer weights w, the positive exponent c_w and the negative exponent \tilde c_w can differ (the paper's Example 7 shows 1.914… versus 1.345…), so the exact optimal exponent for arbitrary weights is not settled by this paper; the gap marks a concrete open problem.
  • The authors report that they could not find earlier results for the d≥2 cases; that is a search statement rather than a proof of novelty, so an independent check of the literature would be needed to confirm priority.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 4 minor

Summary. The paper resolves an Erdős–Graham problem by showing that, for a strictly increasing sequence of positive integers with limsup a_n^{1/φ^n} = ∞, the series ∑ 1/(a_n a_{n+1}) is irrational, where φ is the golden ratio. The main positive result is Theorem 3, a weighted d-dimensional generalization of Erdős's classical criterion, from which the d=2 case follows. A complementary negative result, Theorem 5, constructs strictly increasing integer sequences with lim a_n^{1/\tilde c_w^n} = C for which the sum is rational, establishing sharpness in several cases. The proof is elementary and self-contained, using Mahler's criterion, a Borel-type peak lemma, dyadic-block estimates, and an interval-covering construction for the counterexamples.

Significance. If correct, this settles a 1980 problem of Erdős and Graham and identifies the optimal double-exponential growth threshold, replacing the exponent 2 by the golden ratio. The positive result is broad and the negative result is a strong complement. The paper is largely self-contained and the main analytic lemmas (Lemma 10, Lemma 12, Lemma 14) are carefully argued; the proof of Proposition 11 is detailed and the reduction from the abstract's limsup condition to Theorem 3 is explicitly addressed in Remark 4(3). The negative construction in Lemma 14 is a nice interval-covering argument. However, one case split in the proof of Proposition 11 is incomplete, as detailed below; the gap is local and repairable.

major comments (1)
  1. [§3, Proposition 11, Case (C)] The case analysis contains a false dichotomy. The text states: 'Neither (A) nor (B) holds. This means ... a_n < e^n for infinitely many n.' But failure of Case (B) ('a_n ≥ e^n for every n') only gives at least one violation, not infinitely many. A concrete sequence satisfying the hypotheses of Theorem 3 is: fix d=2, w=(1,1), c=φ, choose N0 large, set a_n = n for n<N0 and a_n = floor(e^{n φ^n}) for n≥N0. This sequence is strictly increasing, satisfies a_n ≥ n eventually and lim a_n^{1/φ^n}=∞. It violates (B) only finitely, and for it Case (A) also fails: log x_{N+1} ~ (N+1)φ^{N+1}, while log D_N^M ~ M N φ^{N+1}/(φ-1) with M=4, so D_N^M ≫ x_{N+1}. The proof then enters Case (C), but P(R) is eventually constant, breaking the subsequent estimates. The fix is to replace Case (B) by 'a_n ≥ e^n for all sufficiently large n', or to pass to a tail of the sequence; the Case (B) argument works unde
minor comments (4)
  1. [§2, Theorem 2(1) and Remark 4(3)] The abstract advertises limsup a_n^{1/φ^n}=∞ for strictly increasing sequences, while Theorem 2(1) states lim a_n^{1/ψ^n}=∞ for non-decreasing sequences. The proof of the abstract's claim in Remark 4(3) uses strict increase to assert a_n ≥ n; this is false for non-decreasing sequences, though under the lim hypothesis it holds eventually. Please align the statements or explicitly note the eventual version.
  2. [§4, Lemma 14 proof] The notation in the displayed comparison of terms is garbled in places (e.g., 'a_w0_n · · · a_wN−n+1_N+1'). Please rewrite with standard exponents and ensure the range of summation indices is clear.
  3. [§4, Proof of Theorem 5] When computing the limit in (4.3), the floors in β_n and γ_n are absorbed into O(n); please state this explicitly to avoid confusion.
  4. [§1, Declaration of AI usage] The AI usage declaration is transparent, but the journal may have specific policies on AI-assisted authorship and verification; please ensure compliance.

Circularity Check

0 steps flagged

No significant circularity; the main d>=2 results derive from self-contained analytic lemmas and fully reproved classical criteria, with only a non-load-bearing self-citation.

full rationale

The paper's central derivation is non-circular. Theorem 2(1) is obtained by specializing Theorem 3, whose proof rests on Lemma 8 (Mahler's criterion, proved inline), Lemma 9 (proved inline), Lemma 10 (proved inline), and the paper's own Lemma 12 and Proposition 11; no parameter appearing in the conclusion is fitted from the conclusion. The only self-citation is [18] (Kovac-Tao), used in the preamble to Lemma 14 as a pointer to 'somewhat similar constructions' and in a remark on the well-known Sylvester sequence; neither occurrence supplies a load-bearing step in the proof of Theorem 2, 3, or 5. The construction in Theorem 5 uses Lemma 14's interval-covering argument, which is an existence result from explicit bounds, not a renaming or a fitted prediction. The skeptic's concern about 'monotonically increasing' versus 'strictly increasing' is a potential correctness gap in the reduction (a_n >= n), not a circularity, since the theorem would still have to be checked against independent hypotheses; it does not make the conclusion equal to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted to data; the constants c_w and c̃_w are uniquely defined roots of polynomials determined by the weights w. The proof relies on classical irrationality criteria and elementary estimates. The integer-valuedness of the sequences is the key domain assumption; the analytic core (Proposition 11) does not use it.

axioms (5)
  • standard math Mahler/Fourier irrationality criterion (Lemma 8): if S is rational and D_N times the N-th truncated sum is an integer, then liminf_{N→∞} D_N (S - partial_N) > 0.
    Proved in the paper; a classical result attributed to Fourier/Mahler. It is the engine of the positive direction.
  • standard math Borel peak-index trick (Lemma 9): if δ_n is summable and limsup μ_n = ∞, then there are infinitely many indices m with μ_{m+1} > (1+δ_m) max_{k≤m} μ_k.
    Proved in the paper and imported from Erdős [4]. Used to select the record indices N ∈ P in Proposition 11.
  • standard math Dyadic-block tail estimates (Lemma 10).
    Proved in the paper using elementary estimates; gives the tail bounds used in Cases A, B, and C of Proposition 11.
  • standard math Descartes' rule of signs to prove uniqueness and positivity of c_w and c̃_w.
    Used in Remark 4(1) and Theorem 5 to ensure the growth constants are well-defined and lie in (1,∞).
  • domain assumption a_n and b_n are positive integers; a_n is monotonically increasing.
    Integer-valuedness is required for D_N = ∏ a_k^W to make the truncated sum an integer in Mahler's criterion; monotonicity forces a_n ≥ n, giving the polynomial growth ∏ a ≥ n^{1+τ} for the d=2 reduction in Remark 4(3).

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Cite this review

Pith. "Pith review of Irrationality of rapidly converging series: a problem of Erd\H{o}s and Graham." pith.science (2026). https://pith.science/paper/YYW2XEEV

@misc{pith2026260121442,
  author       = {Pith},
  title        = {Pith review of: Irrationality of rapidly converging series: a problem of Erd\Hos and Graham},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YYW2XEEV}},
  note         = {Machine review of arXiv:2601.21442}
}
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read the original abstract

Answering a question of Erd\H{o}s and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/\phi^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $\phi$ denotes the golden ratio. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

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Reference graph

Works this paper leans on

22 extracted references · cited by 3 Pith papers

  1. [1]

    T. F. Bloom,Erd˝ os problem #1051, https://www.erdosproblems.com/1051, accessed 2026- 01-11

    2026

  2. [2]

    Cantor, ¨Uber die einfachen zahlensysteme, Zeitschrift f¨ ur Mathematik und Physik14 (1869), 121–128

    G. Cantor, ¨Uber die einfachen zahlensysteme, Zeitschrift f¨ ur Mathematik und Physik14 (1869), 121–128

  3. [3]

    de Stainville,M´ elanges d’analyse alg´ ebrique et de g´ eom´ etrie, Courcier, Paris, 1815

    J. de Stainville,M´ elanges d’analyse alg´ ebrique et de g´ eom´ etrie, Courcier, Paris, 1815

  4. [4]

    Erd˝ os,Some problems and results on the irrationality of the sum of infinite series, J

    P. Erd˝ os,Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci.10(1975), 1–7. MR 539489

    1975

  5. [5]

    Erd˝ os,On the irrationality of certain series: problems and results, New advances in tran- scendence theory (Durham, 1986), Cambridge Univ

    P. Erd˝ os,On the irrationality of certain series: problems and results, New advances in tran- scendence theory (Durham, 1986), Cambridge Univ. Press, Cambridge, 1988, pp. 102–109. MR 971997

    1986

  6. [6]

    Erd˝ os and R

    P. Erd˝ os and R. L. Graham,Old and new problems and results in combinatorial num- ber theory, Monographies de L’Enseignement Math´ ematique [Monographs of L’Enseignement Math´ ematique], vol. 28, Universit´ e de Gen` eve, L’Enseignement Math´ ematique, Geneva, 1980. MR 592420

    1980

  7. [7]

    Erd˝ os and E

    P. Erd˝ os and E. G. Straus,On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.)27(1964), 129–133. MR 175848

    1964

  8. [8]

    Erd˝ os and E

    P. Erd˝ os and E. G. Straus,Some number theoretic results, Pacific J. Math.36(1971), 635–646. MR 281698

    1971

  9. [9]

    Erd˝ os and E

    P. Erd˝ os and E. G. Straus,On the irrationality of certain series, Pacific J. Math.55(1974), 85–92. MR 366823

    1974

  10. [10]

    T. Feng, T. H. Trinh, G. Bingham, Y. Chervonyi, J. Jung, D. Hwang, J. Lee, C. Pagano, S. Kim, F. Pasqualotto, S. Gukov, J. Kim, K. Hou, J. Lee, G. Ghiasi, Y. Tay, Y. Li, C. Kuang, Y. Liu, H. Lin, E. Liu, N. Nayakanti, X. Yang, H. Cheng, K. Kavukcuoglu, Q. V. Le, and T. Luong,Towards Autonomous Mathematics Research, preprint, 2026

    2026

  11. [11]

    Google DeepMind, Aletheia (GitHub), 2026, Available athttps://github.com/ google-deepmind/superhuman/tree/main/aletheia, Accessed: 2026-01-29

    2026

  12. [12]

    Hanˇ cl,Expression of real numbers with the help of infinite series, Acta Arith.59(1991), no

    J. Hanˇ cl,Expression of real numbers with the help of infinite series, Acta Arith.59(1991), no. 2, 97–104. MR 1133951

    1991

  13. [13]

    Hanˇ cl and F

    J. Hanˇ cl and F. Luca,Irrationality of infinite series, Mediterr. J. Math.16(2019), no. 4, Paper No. 89, 13. MR 3969262

    2019

  14. [14]

    Hanˇ cl and R

    J. Hanˇ cl and R. Nair,On the irrationality of infinite series of reciprocals of square roots, Rocky Mountain J. Math.47(2017), no. 5, 1525–1538. MR 3705763

    2017

  15. [15]

    Hanˇ cl and S

    J. Hanˇ cl and S. Sobkov´ a,A general criterion for linearly unrelated sequences, Tsukuba J. Math.27(2003), no. 2, 341–357. MR 2025732

    2003

  16. [16]

    Hanˇ cl and R

    J. Hanˇ cl and R. Tijdeman,On the irrationality of Cantor and Ahmes series, Publ. Math. Debrecen65(2004), no. 3-4, 371–380. MR 2107954

    2004

  17. [17]

    Hanˇ cl and R

    J. Hanˇ cl and R. Tijdeman,On the irrationality of Cantor series, J. Reine Angew. Math.571 (2004), 145–158. MR 2070147

    2004

  18. [18]

    Kovaˇ c and T

    V. Kovaˇ c and T. Tao,On several irrationality problems for Ahmes series, Acta Math. Hungar. 175(2025), no. 2, 572–608. MR 4915145

    2025

  19. [19]

    Mahler,Lectures on transcendental numbers, Lecture Notes in Mathematics, vol

    K. Mahler,Lectures on transcendental numbers, Lecture Notes in Mathematics, vol. Vol. 546, Springer-Verlag, Berlin-New York, 1976. MR 491533

    1976

  20. [20]

    Nishioka,Mahler functions and transcendence, Lecture Notes in Mathematics, vol

    K. Nishioka,Mahler functions and transcendence, Lecture Notes in Mathematics, vol. 1631, Springer-Verlag, Berlin, 1996. MR 1439966 16 K. BARRETO, J. KANG, S. KIM, V. KOV A ˇC, AND S. ZHANG

    1996

  21. [21]

    Oppenheim,Criteria for irrationality of certain classes of numbers, Amer

    A. Oppenheim,Criteria for irrationality of certain classes of numbers, Amer. Math. Monthly 61(1954), 235–241. MR 61140

    1954

  22. [22]

    Tijdeman and P

    R. Tijdeman and P. Yuan,On the rationality of Cantor and Ahmes series, Indag. Math. (N.S.)13(2002), no. 3, 407–418. MR 2057056 Queens’ College, University of Cambridge, Cambridge, CB3 9ET, United Kingdom Email address:kb799@cam.ac.uk Department of Mathematical Science, Seoul National University, Seoul, 08826, Ko- rea Email address:carl011226@gmail.com Sch...

    2002

This paper was first reviewed by deepseek-v4-flash on August 3, 2026.