Reciprocal Sums and the Sylvester Recurrence

Will Cook

Short paper

Reciprocal Sums and the Sylvester Recurrence

Erdős #243 By Will Cook September 2026 Authorship, AI use and citation

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Summary. For strictly increasing positive integers with $a_{n}^{2} / a_{n + 1} = 1 + 3 / n + o (n^{- 3})$ , the reciprocal sum is irrational. The argument extracts a cubic polynomial from an integer tail numerator, restricts its coefficients using a square in a cubic field, and excludes the remaining cases modulo seven. Under rationality and the quadratic growth limit, separate criteria using bounded numerator increases and weighted first crossings force the eventual Sylvester recurrence. The unrestricted Erdős–Graham question remains open.

This paper covers cubic-rate irrationality, nonintegral regular-rate exclusion and sufficient tail criteria for the Sylvester recurrence; the unrestricted Erdős–Graham question remains open.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open.

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References

P. Erdős and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.) 27 (1964), 129–133. MR 175848.
D. Duverney, Irrationality of fast converging series of rational numbers, J. Math. Sci. Univ. Tokyo 8 (2001), 275–316. MR 1837165.
C. Badea, A theorem on irrationality of infinite series and applications, Acta Arith. 63 (1993), no. 4, 313–323, doi:10.4064/aa-63-4-313-323.
R. Tijdeman and P. Yuan, On the rationality of Cantor and Ahmes series, Indag. Math. (N.S.) 13 (2002), no. 3, 407–418, doi:10.1016/S0019-3577(02)80018-0.
P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monogr. Enseign. Math. 28, Geneva, 1980, p. 64.
P. Erdős, On the irrationality of certain series: problems and results, in A. Baker (ed.), New Advances in Transcendence Theory, Cambridge UP, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.
I. O. Bado, Prime-Support Rigidity and Primitive Pseudo-Greedy Dynamics: Partial Progress on Erdős Problem #243, preprint posted September 2026, doi:10.13140/RG.2.2.36612.08325.
P. White with Claude (Anthropic), Erdős #243: working report, Erdős Problem a Day, page dated 12 August 2026. AI-assisted, unrefereed working report.
P. Stevenhagen and H. W. Lenstra, Jr., Chebotarëv and his density theorem, Math. Intelligencer 18 (1996), no. 2, 26–37, doi:10.1007/BF03027290. Page references are to the linked author version dated 23 March 1995.
J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, Integers 26 (2026), Paper No. A28, 17 pp., doi:10.5281/zenodo.18714404. Locators refer to this published version.
T. F. Bloom, Erdős Problem #243, accessed 28 July 2026.
P. Erdős and E. G. Straus, On the irrationality of certain series, Pacific J. Math. 55 (1974), no. 1, 85–92.
J. Hančl and R. Tijdeman, On the irrationality of polynomial Cantor series, Acta Arith. 133 (2008), no. 1, 37–52, doi:10.4064/aa133-1-3. Locators refer to the published version.
D. Duverney, T. Kurosawa and I. Shiokawa, Irrationality exponents of certain fast converging series of rational numbers, Tsukuba J. Math. 44 (2020), no. 2, 235–250, doi:10.21099/tkbjm/20204402235. Theorem locators follow the linked 14-page author version.
T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series, Colloq. Math. 179 (2025), 55–68, doi:10.4064/cm9628-5-2025. Theorem locators follow arXiv:2504.18712v1.
A. Koutsoukou-Argyraki and W. Li, Irrationality Criteria for Series by Erdős and Straus, Archive of Formal Proofs, 12 May 2020. An Isabelle/HOL formalisation; the archive entry identifies the results formalised.
National Institute of Standards and Technology, Digital Library of Mathematical Functions, §5.11(iii), formula 5.11.12, accessed 16 September 2026.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

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[ref-1] P. Erdős and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.) 27 (1964), 129–133. MR 175848.

[ref-2] D. Duverney, Irrationality of fast converging series of rational numbers, J. Math. Sci. Univ. Tokyo 8 (2001), 275–316. MR 1837165.

[ref-3] C. Badea, A theorem on irrationality of infinite series and applications, Acta Arith. 63 (1993), no. 4, 313–323, doi:10.4064/aa-63-4-313-323.

[ref-4] R. Tijdeman and P. Yuan, On the rationality of Cantor and Ahmes series, Indag. Math. (N.S.) 13 (2002), no. 3, 407–418, doi:10.1016/S0019-3577(02)80018-0.

[ref-5] P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monogr. Enseign. Math. 28, Geneva, 1980, p. 64.

[ref-6] P. Erdős, On the irrationality of certain series: problems and results, in A. Baker (ed.), New Advances in Transcendence Theory, Cambridge UP, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.

[ref-7] I. O. Bado, Prime-Support Rigidity and Primitive Pseudo-Greedy Dynamics: Partial Progress on Erdős Problem #243, preprint posted September 2026, doi:10.13140/RG.2.2.36612.08325.

[ref-8] P. White with Claude (Anthropic), Erdős #243: working report, Erdős Problem a Day, page dated 12 August 2026. AI-assisted, unrefereed working report.

[ref-9] P. Stevenhagen and H. W. Lenstra, Jr., Chebotarëv and his density theorem, Math. Intelligencer 18 (1996), no. 2, 26–37, doi:10.1007/BF03027290. Page references are to the linked author version dated 23 March 1995.

[ref-10] J. Koizumi, Irrationality of the reciprocal sum of doubly exponential sequences, Integers 26 (2026), Paper No. A28, 17 pp., doi:10.5281/zenodo.18714404. Locators refer to this published version.

[ref-11] T. F. Bloom, Erdős Problem #243, accessed 28 July 2026.

[ref-12] P. Erdős and E. G. Straus, On the irrationality of certain series, Pacific J. Math. 55 (1974), no. 1, 85–92.

[ref-13] J. Hančl and R. Tijdeman, On the irrationality of polynomial Cantor series, Acta Arith. 133 (2008), no. 1, 37–52, doi:10.4064/aa133-1-3. Locators refer to the published version.

[ref-14] D. Duverney, T. Kurosawa and I. Shiokawa, Irrationality exponents of certain fast converging series of rational numbers, Tsukuba J. Math. 44 (2020), no. 2, 235–250, doi:10.21099/tkbjm/20204402235. Theorem locators follow the linked 14-page author version.

[ref-15] T. Crmarić and V. Kovač, On the irrationality of certain super-polynomially decaying series, Colloq. Math. 179 (2025), 55–68, doi:10.4064/cm9628-5-2025. Theorem locators follow arXiv:2504.18712v1.

[ref-16] A. Koutsoukou-Argyraki and W. Li, Irrationality Criteria for Series by Erdős and Straus, Archive of Formal Proofs, 12 May 2020. An Isabelle/HOL formalisation; the archive entry identifies the results formalised.

[ref-17] National Institute of Standards and Technology, Digital Library of Mathematical Functions, §5.11(iii), formula 5.11.12, accessed 16 September 2026.