Reading Eight Erdős Problems Together

Will Cook

Cross-problem paper

Reading Eight Erdős Problems Together

Eight Erdős problems read together By Will Cook 22 September 2026 Authorship, AI use and citation

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Précis. One account of the mathematics developed across the programmes: an exact tail-capacity criterion under eventual congruences, its sharp factorial support-gap threshold, rational and irrational Lambert subsums across bases, and the limits of finite greedy tests and other irrationality methods. Full arguments, exact counterexamples, unsuccessful routes and attribution are retained together. The principal proofs are ordinary mathematics; cited Lean ingredients have their own stated scope. Historical novelty and independent expert review are not established.

This paper owns the synthesis exposition and ordinary proofs across the covered problems: capacity and congruence constructions, Lambert subsums, rational-point counts, method limits and their research record.

It is not authority for a solution to any original Erdős target, historical novelty, independent expert review, or a full Lean proof of the analytic capacity criterion or Lambert-chain theorem.

Start here, by the editorial route

In this paper

References

T. H. Lê, Intersective polynomials and the primes, arXiv:0910.1880v1.
A. Rice, Sárközy’s theorem for $P$ -intersective polynomials, arXiv:1111.6559.
B. Mishra, Polynomials consisting of quadratic factors with roots modulo any positive integer, Amer. Math. Monthly 129 (2022), 178–182, arXiv:2102.08379.
S. Akiyama and H. Kaneko, Multiplicative analogue of Markoff–Lagrange spectrum and Pisot numbers, Adv. Math. 380 (2021), 107547, p. 3, reporting the theorem of A. Dubickas, arXiv:1911.06170v6.
J.-P. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: the Sharkovskiı̆ ordering, Ergodic Theory Dynam. Systems 29 (2009), 1055–1074, doi:10.1017/S0143385708000746.
P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monogr. Enseign. Math. 28, Geneva, 1980.
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.
P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969).
H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.
Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013).
John A. Fridy, Generalized bases for the real numbers. The Fibonacci Quarterly 4 (1966), no. 3, 193–201.
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.
Vjekoslav Kovač and Terence Tao, On several irrationality problems for Ahmes series. Acta Mathematica Hungarica 175 (2025), 572–608, doi:10.1007/s10474-025-01528-0; arXiv:2406.17593v4. Statement and section numbers refer to arXiv version 4.
W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, doi:10.4064/aa111-2-4.
T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026).
Steve Fan, Comment on Erdős Problem #269, thread 269, post 7218 (2026), source. 26 June 2026.
K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342–367.
B. Adamczewski and C. Faverjon, Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques, arXiv:1508.07158v2 (2016), Theorem 1.1, p. 3.
Ku. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Math. 1631, Springer, Berlin, 1996.
W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551.
H. P. Schlickewei, Die $p$ -adische Verallgemeinerung des Satzes von Thue–Siegel–Roth–Schmidt, J. Reine Angew. Math. 288 (1976), 86–105.
Yu. F. Bilu, The many faces of the subspace theorem [after Adamczewski, Bugeaud, Corvaja, Zannier…], Séminaire Bourbaki, Vol. 2006/2007, Exp. No. 967, Astérisque 317 (2008), 1–38; arXiv:0907.2098. Theorem 2.3 is the statement used here.
P. Corvaja and U. Zannier, Some new applications of the subspace theorem, Compositio Math. 131 (2002), 319–340, doi:10.1023/A:1015594913393.
J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables III, Bull. Austral. Math. Soc. 16 (1977), 15–47, doi:10.1017/S0004972700022978.
K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20.
D. Boes, R. Darst and P. Erdős, Fat, symmetric, irrational Cantor sets, Amer. Math. Monthly 88 (1981), 340–341.
A. Rahm, N. Solomon, T. Trauthwein and B. Weiss, The distribution of rational numbers on Cantor’s middle thirds set, arXiv:1909.01198.
S. Chow, P. Varjú and H. Yu, Counting rationals and diophantine approximation in missing-digit Cantor sets, arXiv:2402.18395.
W. Cook, Two incomparable denominator exclusions for $\sum_{n \geq 2} (n! - 1)^{- 1}$ , Erdős Problem Notes, 2026.
W. Cook, Bounded increments and rational reciprocal sums, Erdős Problem Notes, 2026.
W. Cook, Bases and integral relations for the $k$ -kernel of Euler’s totient, Erdős Problem Notes, 2026.
W. Cook, Sparse congruence-preserving perturbations of dyadic series, Erdős Problem Notes, 2026.
W. Cook, Weighted support criteria for reciprocal Mersenne subseries, Erdős Problem Notes, 2026.
W. Cook, No finite separable representation at three prime generators, Erdős Problem Notes, 2026.
W. Cook, Zudilin’s forms at rational bases and the exact normalised Hankel order, Erdős Problem Notes, 2026.
W. Cook, Choices, contraction and rational membership, exact computation with saved outputs in research/experiments/choices_contraction/ of the repository, 2026.
P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.
Wouter van Doorn and Vjekoslav Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval. Acta Arithmetica 223 (2026), 275–295, doi:10.4064/aa251001-13-1; arXiv:2509.24971v3. Statement numbers refer to arXiv version 3.
D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint.
Boris Adamczewski and Colin Faverjon, Mahler’s method in several variables and finite automata, Annals of Mathematics 204, no. 2 (2026), 455–533, doi:10.4007/annals.2026.204.2.1. Locators refer to the 68-page author manuscript.
D. Airey, B. Mance and J. Vandehey, Normality preserving operations for Cantor series expansions and associated fractals. II, New York J. Math. 21 (2015), 1311–1326. https://nyjm.albany.edu/j/2015/21-60v.pdf.
M. Waldschmidt, Integer-valued functions, Hurwitz functions, and related topics: a survey, in K. Kurşungöz and A. Zeytin (eds.), Number Theory: Proceedings of the Journées Arithmétiques 2019, De Gruyter, 2022, pp. 61–82. https://doi.org/10.1515/9783110761115-005.
H. Wegmann, Die Hausdorffsche Dimension von Mengen reeller Zahlen, die durch Zifferneigenschaften einer Cantorentwicklung charakterisiert sind, Czechoslovak Math. J. 18 (1968), 622–632. https://dml.cz/handle/10338.dmlcz/100861.
J.-H. Evertse and R. Ferretti, A further improvement of the quantitative Subspace Theorem, Ann. of Math. 177 (2013), 513–590. Normalizations refer to §1.1, (1.1), and §2.1 of the 3 May 2012 author version.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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[ref-1] T. H. Lê, Intersective polynomials and the primes, arXiv:0910.1880v1.

[ref-2] A. Rice, Sárközy’s theorem for $P$ -intersective polynomials, arXiv:1111.6559.

[ref-3] B. Mishra, Polynomials consisting of quadratic factors with roots modulo any positive integer, Amer. Math. Monthly 129 (2022), 178–182, arXiv:2102.08379.

[ref-4] S. Akiyama and H. Kaneko, Multiplicative analogue of Markoff–Lagrange spectrum and Pisot numbers, Adv. Math. 380 (2021), 107547, p. 3, reporting the theorem of A. Dubickas, arXiv:1911.06170v6.

[ref-5] J.-P. Allouche, M. Clarke and N. Sidorov, Periodic unique beta-expansions: the Sharkovskiı̆ ordering, Ergodic Theory Dynam. Systems 29 (2009), 1055–1074, doi:10.1017/S0143385708000746.

[ref-6] P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monogr. Enseign. Math. 28, Geneva, 1980.

[ref-7] 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-8] P. Erdős, On the irrationality of certain series, Math. Student 36 (1968), 222–226 (issued 1969).

[ref-9] H. Hornich, Über beliebige Teilsummen absolut konvergenter Reihen, Monatshefte für Mathematik und Physik 49 (1941), 316–320. doi:10.1007/BF01707309.

[ref-10] Z. Nitecki, Subsum sets: intervals, Cantor sets, and Cantorvals, arXiv:1106.3779v2 (2013).

[ref-11] John A. Fridy, Generalized bases for the real numbers. The Fibonacci Quarterly 4 (1966), no. 3, 193–201.

[ref-12] 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-13] Vjekoslav Kovač and Terence Tao, On several irrationality problems for Ahmes series. Acta Mathematica Hungarica 175 (2025), 572–608, doi:10.1007/s10474-025-01528-0; arXiv:2406.17593v4. Statement and section numbers refer to arXiv version 4.

[ref-14] W. Zudilin, Heine’s basic transform and a permutation group for $q$ -harmonic series, Acta Arith. 111 (2004), no. 2, 153–164, doi:10.4064/aa111-2-4.

[ref-15] T. Tao and J. Teräväinen, Quantitative correlations and some problems on prime factors of consecutive integers, arXiv:2512.01739v2 (submitted December 2025, revised April 2026).

[ref-16] Steve Fan, Comment on Erdős Problem #269, thread 269, post 7218 (2026), source. 26 June 2026.

[ref-17] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342–367.

[ref-18] B. Adamczewski and C. Faverjon, Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques, arXiv:1508.07158v2 (2016), Theorem 1.1, p. 3.

[ref-19] Ku. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Math. 1631, Springer, Berlin, 1996.

[ref-20] W. M. Schmidt, Norm form equations, Ann. of Math. (2) 96 (1972), 526–551.

[ref-21] H. P. Schlickewei, Die $p$ -adische Verallgemeinerung des Satzes von Thue–Siegel–Roth–Schmidt, J. Reine Angew. Math. 288 (1976), 86–105.

[ref-22] Yu. F. Bilu, The many faces of the subspace theorem [after Adamczewski, Bugeaud, Corvaja, Zannier…], Séminaire Bourbaki, Vol. 2006/2007, Exp. No. 967, Astérisque 317 (2008), 1–38; arXiv:0907.2098. Theorem 2.3 is the statement used here.

[ref-23] P. Corvaja and U. Zannier, Some new applications of the subspace theorem, Compositio Math. 131 (2002), 319–340, doi:10.1023/A:1015594913393.

[ref-24] J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables III, Bull. Austral. Math. Soc. 16 (1977), 15–47, doi:10.1017/S0004972700022978.

[ref-25] K. F. Roth, Rational approximations to algebraic numbers, Mathematika 2 (1955), 1–20.

[ref-26] D. Boes, R. Darst and P. Erdős, Fat, symmetric, irrational Cantor sets, Amer. Math. Monthly 88 (1981), 340–341.

[ref-27] A. Rahm, N. Solomon, T. Trauthwein and B. Weiss, The distribution of rational numbers on Cantor’s middle thirds set, arXiv:1909.01198.

[ref-28] S. Chow, P. Varjú and H. Yu, Counting rationals and diophantine approximation in missing-digit Cantor sets, arXiv:2402.18395.

[ref-29] W. Cook, Two incomparable denominator exclusions for $\sum_{n \geq 2} (n! - 1)^{- 1}$ , Erdős Problem Notes, 2026.

[ref-30] W. Cook, Bounded increments and rational reciprocal sums, Erdős Problem Notes, 2026.

[ref-31] W. Cook, Bases and integral relations for the $k$ -kernel of Euler’s totient, Erdős Problem Notes, 2026.

[ref-32] W. Cook, Sparse congruence-preserving perturbations of dyadic series, Erdős Problem Notes, 2026.

[ref-33] W. Cook, Weighted support criteria for reciprocal Mersenne subseries, Erdős Problem Notes, 2026.

[ref-34] W. Cook, No finite separable representation at three prime generators, Erdős Problem Notes, 2026.

[ref-35] W. Cook, Zudilin’s forms at rational bases and the exact normalised Hankel order, Erdős Problem Notes, 2026.

[ref-36] W. Cook, Choices, contraction and rational membership, exact computation with saved outputs in research/experiments/choices_contraction/ of the repository, 2026.

[ref-37] P. Erdős, On arithmetical properties of Lambert series, J. Indian Math. Soc. 12 (1948), 63–66.

[ref-38] Wouter van Doorn and Vjekoslav Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval. Acta Arithmetica 223 (2026), 275–295, doi:10.4064/aa251001-13-1; arXiv:2509.24971v3. Statement numbers refer to arXiv version 3.

[ref-39] D. Duverney and Y. Tachiya, Refinement of the Chowla–Erdős method and linear independence of certain Lambert series, Forum Math. 31 (2019), no. 6, 1557–1566, DOI. Page numbers refer to the linked author preprint.

[ref-40] Boris Adamczewski and Colin Faverjon, Mahler’s method in several variables and finite automata, Annals of Mathematics 204, no. 2 (2026), 455–533, doi:10.4007/annals.2026.204.2.1. Locators refer to the 68-page author manuscript.

[ref-41] D. Airey, B. Mance and J. Vandehey, Normality preserving operations for Cantor series expansions and associated fractals. II, New York J. Math. 21 (2015), 1311–1326. https://nyjm.albany.edu/j/2015/21-60v.pdf.

[ref-42] M. Waldschmidt, Integer-valued functions, Hurwitz functions, and related topics: a survey, in K. Kurşungöz and A. Zeytin (eds.), Number Theory: Proceedings of the Journées Arithmétiques 2019, De Gruyter, 2022, pp. 61–82. https://doi.org/10.1515/9783110761115-005.

[ref-43] H. Wegmann, Die Hausdorffsche Dimension von Mengen reeller Zahlen, die durch Zifferneigenschaften einer Cantorentwicklung charakterisiert sind, Czechoslovak Math. J. 18 (1968), 622–632. https://dml.cz/handle/10338.dmlcz/100861.

[ref-44] J.-H. Evertse and R. Ferretti, A further improvement of the quantitative Subspace Theorem, Ann. of Math. 177 (2013), 513–590. Normalizations refer to §1.1, (1.1), and §2.1 of the 3 May 2012 author version.