Koizumi 2025 irrationality reciprocal sum doubly exponential sequences

../

conjecture_6: Koizumi's conjecture that for a positive rational r whose pseudo-greedy expansion has gap sequence tending to 0, the gap is 0 from some index on; with the definitions of the expansion and its gap sequence, the computer check to 10^5 and the heuristic of Remark 17.

corollary_2: A sequence of positive integers with every ratio a_n^2/a_{n+1} between 2/3 and 4/3 and reciprocal sum exactly 1 is Sylvester's sequence 2, 3, 7, 43, ..., which settles the question of Problem 243 for such sequences.

corollary_20: Koizumi's recovery, through the gap sequence of the pseudo-greedy expansion, of the Erdős-Straus and Badea conditions under which a sequence with a_n^2/a_{n+1} -> 1 and rational reciprocal sum eventually satisfies a_{n+1} = a_n^2 - a_n + 1; with Proposition 19, the gap-sequence form.

remark_22: Koizumi's remark that if the Sylvester-recurrence question of Problem 243 has an affirmative answer, the exceptional set of Theorem 4 consists of the transcendental numbers c(m)^{2^{-N}}, so every algebraic alpha > 1, including alpha = 2 of Problem 263, is not exceptional.

theorem_1: If every a_n^2/a_{n+1} lies within 1/3 of a fixed beta >= 0 and the reciprocal sum r is finite, each term with a_n >= 8(beta+1/3)^2 is the nearest integer to beta plus the reciprocal of the remainder of r.

theorem_16: Koizumi's Conjecture 6 on pseudo-greedy expansions of rationals holds if and only if every positive-integer sequence with a_n^2/a_{n+1} -> 1 and rational reciprocal sum eventually satisfies a_{n+1} = a_n^2 - a_n + 1, the question of Problem 243.

theorem_4: For all real alpha > 1 outside a countable set, every sequence of positive integers asymptotic to alpha^{2^n} has irrational reciprocal sum; the base alpha = 2 of Problem 263 is not decided.

---

Junnosuke Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences. arXiv preprint (2025). arXiv:2504.05933.

The copy read for this card is arXiv:2504.05933v1 (8 April 2025, 14 pages; the only arXiv version listed on 2026-09-18). The paper is published as Integers 26 (2026), paper A28, 17 pages (received 10/9/25, accepted 1/15/26, published 2/20/26), compared below (Editions); the labels and pages named in this digest and on the problem pages are the preprint's. Koizumi proves that if a sequence of positive integers has finite reciprocal sum r and its ratios a_n^2/a_{n+1} all lie within 1/3 of a fixed real beta >= 0, then each term with a_n >= 8(beta+1/3)^2 is the integer nearest to beta + 1/(r - 1/a_1 - ... - 1/a_{n-1}) (Theorem 1, p. 2), so that for instance the Sylvester sequence is the only sequence of positive integers with every a_n^2/a_{n+1} in [2/3,4/3] and reciprocal sum 1 (Corollary 2). Koizumi deduces that for every real alpha > 1 outside a countable set, every sequence of positive integers asymptotic to alpha^{2^n} has irrational reciprocal sum (Theorem 4, p. 3). Koizumi also proves that an open Erdos-Graham question (Question 5, p. 3) has an affirmative answer exactly when Conjecture 6 on the pseudo-greedy expansion (pp. 3 and 7-9) holds (Theorem 16, p. 10), and gives a heuristic argument for that conjecture (Remark 17, p. 11). The method is a rigidity argument on greedy-type approximations of the reciprocal sum. For problem 282 it is the 2025 paper Vjekoslav Kovac's site comment cited for the connection between irrationality and greedy-type approximation; Section 2 (p. 7) defines the odd greedy expansion (the smallest odd denominator not below the reciprocal of the remainder) and records: "It is an open problem whether the odd greedy expansion of a positive rational number with odd denominator terminates in finite steps", citing Guy's problem book, and p. 9 notes that Conjecture 6 "resembles the termination problem of the odd greedy expansion". Read status: claims checked. Theorem 1, Corollaries 2 and 3, Theorem 4, Question 5, Conjecture 6, Theorem 16 and the odd-greedy passages were read clause by clause on the page images (pp. 2-3, 7, 9 and 10), and the statements of Definitions 7 and 9, Proposition 19, Corollary 20 and Remarks 17, 21 and 22 on pp. 7-8 and 11-13; no proof was checked. Each result page records its own read depth.

Source: https://arxiv.org/abs/2504.05933.

Editions. The arXiv v1 text has 14 pages with a text layer. The alternate journal PDF is the published version, Integers 26 (2026), paper A28 (17 pages, numbered 1--17; DOI 10.5281/zenodo.18714404 printed on p. 1). Its provenance: 334,539 bytes, retrieved (the download URL is not recorded). The journal version renumbers the results: preprint Theorem 1 is Theorem 1 (p. 2), Corollary 2 is Corollary 1 (p. 2), Corollary 3 is Corollary 2 (p. 2), Theorem 4 is Theorem 2 (p. 3), Question 5 is Question 1 (p. 3), Conjecture 6 is Conjecture 1 (p. 4), Corollary 10 is Corollary 3 (p. 9), Theorem 16 is Theorem 3 (p. 12), Remark 17 is Remark 1 (p. 13) and Corollary 20 is Corollary 4 (p. 14); the odd-greedy passage of preprint p. 7 is on journal p. 8 (Section 2, checked on the page image) and the "resembles the termination problem of the odd greedy expansion" remark of preprint p. 9 is on journal p. 11. The statements of Theorem 1, Corollaries 1 and 2, Theorem 2, Question 1 and Conjecture 1 and the odd-greedy passage were compared on the page images of both editions and agree apart from the labels, the citation keys (the journal cites Guy's problem book as "[9, p. 88]" and the Erdős--Graham monograph as "[6]" where the preprint has "[Guy81, p. 88]" and "[EG80, p. 64]"), the journal's "n ≫ 1" where the preprint has "n ≫ 0" (each defined as holding for all n >= n_0, for some positive integer n_0: journal p. 4, preprint p. 3) and the journal's "greater than or equal to" where the preprint's odd-greedy passage has "≥"; the remaining text was not compared. For the arXiv preprint, the arXiv record names arXiv's non-exclusive distribution license (arXiv:2504.05933), every other right reserved. For the journal PDF, no notice is printed beyond "DOI: 10.5281/zenodo.18714404" on p. 1, and the Zenodo record of the journal's deposit names the Creative Commons Attribution 4.0 license, license id "cc-by-4.0", with the access right "open" and a file of the same size, 334,539 bytes (https://zenodo.org/api/records/18714404, read 2026-10-02); the journal's home page states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02).

Bears on. #243: the paper's Question 5 is the problem's question (the problem asks for an increasing sequence, which a tail of any sequence meeting Question 5's hypotheses is); Theorem 16 proves it equivalent to Conjecture 6 without settling either, Corollary 2 proves the recurrence for the sequences with every $a_n^2/a_{n+1}$ in $[2/3,4/3]$ and reciprocal sum $1$, and Corollary 20 re-derives the Erdős--Straus and Badea cases. #263: Theorem 4 gives the property of the first question for all $\alpha>1$ outside a countable set and does not decide $\alpha=2$; Remark 22 derives $\alpha=2$ from an affirmative answer to Question 5, a conditional implication. Nothing bears on the second question. #282: Section 2 (p. 7) records the termination of the odd greedy expansion as open, citing Guy's problem book, and p. 9 says Conjecture 6 resembles it; no result of the paper concerns the problem.

Results.

• Theorem 1 (p. 2): with every $a_n^2/a_{n+1}$ within $1/3$ of $\beta\ge0$, each term with $a_n\ge8(\beta+1/3)^2$ is fixed by the reciprocal sum and the earlier terms.
• Corollary 2 (p. 2): Sylvester's sequence is the only one with every $a_n^2/a_{n+1}\in[2/3,4/3]$ and reciprocal sum $1$.
• Theorem 4 (p. 3): $\lfloor\alpha^{2^n}\rfloor$ is a Type 2 irrationality sequence for all but countably many $\alpha>1$.
• Conjecture 6 (p. 3), with Definitions 7 and 9 (pp. 7--8) and Remark 17 (p. 11): vanishing gaps of a rational's pseudo-greedy expansion are eventually zero.
• Theorem 16 (p. 10): Conjecture 6 holds if and only if Question 5 (p. 3) has an affirmative answer.
• Corollary 20 (p. 12), with Proposition 19 (p. 12): the Erdős--Straus and Badea cases of Question 5.
• Remark 22 (p. 13): an affirmative answer to Question 5 would make $2^{2^n}$ a Type 2 irrationality sequence.

Only the edition under an open license is held; the source's other editions are not, since no license on record permits their redistribution, and the card cites the edition it names above.