Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Will Cook

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Prime-Gap Dyadic Series: Perturbations, Exact Criteria and Certificates

Erdős #251 By Will Cook September 2026 Authorship, AI use and citation

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Summary. A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails $T_{N} = \sum_{j \geq 1} g_{N + j} 2^{- j}$ satisfy $T_{N + 1} = 2 T_{N} - g_{N + 1}$ . That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least $2^{39998} > 10^{12040}$ . The bound does not establish irrationality.

This paper covers the complete problem-specific reasoning surface for Erdős #251, including all registered result families and their boundaries.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open.

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References

Paul Erdős, Sur certaines séries à valeur irrationnelle. L’Enseignement Mathématique 4 (1958), 93–100, doi:10.5169/seals-34629.
Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory. Monographies de L’Enseignement Mathématique, L’Enseignement Mathématique, 1980.
Paul Erdős and Carl Pomerance, On the largest prime factors of $n$ and $n + 1$ . Aequationes Mathematicae 17 (1978), 311–321, doi:10.1007/BF01818569.
Paul Erdős, On the irrationality of certain series: problems and results. New Advances in Transcendence Theory, Cambridge University Press, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.
Paul Erdős, Beweis eines Satzes von Tschebyschef. Acta Litterarum ac Scientiarum Szeged 5 (1932), 194–198.
Paul Erdős and Ernst G. Straus, On the irrationality of certain Ahmes series. Journal of the Indian Mathematical Society (N.S.) 27 (1964), 129–133. MR0175848.
Paul Erdős and Ernst G. Straus, On the irrationality of certain series. Pacific Journal of Mathematics 55 (1974), no. 1, 85–92, doi:10.2140/pjm.1974.55.85.
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, Long gaps between primes. Journal of the American Mathematical Society 31 (2018), 65–105, doi:10.1090/jams/876; arXiv:1412.5029v3.
John A. Fridy, Generalized bases for the real numbers. The Fibonacci Quarterly 4 (1966), no. 3, 193–201.
A. Ya. Khinchin, Continued Fractions. University of Chicago Press, 1964.
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.
Vivian Kuperberg, Sums of singular series with large sets and the tail of the distribution of primes. The Quarterly Journal of Mathematics 74 (2023), no. 4, 1457–1479, doi:10.1093/qmath/haad030; arXiv:2210.09775v2. Statement numbers refer to arXiv version 2, 15 June 2023.
Johan Land, A conditional proof of the irrationality of $\sum_{n \geq 1} p_{n} 2^{- n}$ under a uniform Hardy–Littlewood prime-tuples conjecture. Research draft, 5 September 2026. Formalisation material accompanies the draft.
Leonardo de Moura and Sebastian Ullrich, The Lean 4 theorem prover and programming language. Automated Deduction – CADE 28, Lecture Notes in Computer Science, Springer, 2021, pp. 625–635, doi:10.1007/978-3-030-79876-5_37.
The mathlib Community, The Lean mathematical library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, ACM, 2020, pp. 367–381, doi:10.1145/3372885.3373824. Describes a Lean 3-era snapshot; the repository lock identifies the library used by the present Lean 4 sources.
James Maynard, Small gaps between primes. Annals of Mathematics 181 (2015), 383–413, doi:10.4007/annals.2015.181.1.7.
Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007, doi:10.1017/CBO9780511618314.
Jan-Christoph Schlage-Puchta, The irrationality of some number theoretical series. Acta Arithmetica 126 (2007), no. 4, 295–303, doi:10.4064/aa126-4-1; arXiv:1105.1451v1. Published in 2007; arXiv upload is from 2011. Lemma and preprint page locators refer to arXiv version 1.
Ian Short, Ford Circles, Continued Fractions, and Rational Approximation. The American Mathematical Monthly 118 (2011), no. 2, 130–135, doi:10.4169/amer.math.monthly.118.02.130; arXiv:0912.1997v1. Preprint title: Ford circles, continued fractions, and best approximation of the second kind. Preprint theorem locators refer to arXiv version 1.
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.
Yitang Zhang, Bounded gaps between primes. Annals of Mathematics 179 (2014), 1121–1174, doi:10.4007/annals.2014.179.3.7.
Thomas F. Bloom, Erdős Problem #251. 2026. Accessed 6 September 2026.
Erdős Problems contributors, Erdős Problem #251 discussion thread. 2026. Accessed 6 September 2026: Tao comment of 7 October 2025 and Land comments of 6 September 2026.
ChatGPT 5.4 Pro (orchestrated by Vjeko Kovač), On the Erdős problem #251. Unpublished note, Department of Mathematics, University of Zagreb, 2026. Accessed 6 September 2026.
The Formal Conjectures Authors, FormalConjectures.ErdosProblems.251. 2025. Pinned statement source, not a proof of Problem 251; accessed 28 July 2026.
Stefan Ringer, Local gap statistics, telescoping, and normality: a local-pattern approach to Erdős problem 251. Preprint, 11 September 2026. Mutable main-branch TeX consulted 16 September 2026; the cited conditional statements were rechecked 18 September 2026. Formalisation material accompanies the draft.
Tonći Crmarić and Vjekoslav Kovač, On the irrationality of certain super-polynomially decaying series. Colloquium Mathematicum 179 (2025), 55–68, doi:10.4064/cm9628-5-2025; arXiv:2504.18712v1. Lemma 4 is cited using arXiv version 1.
Boris Adamczewski, Michael Drmota and Clemens Müllner, (Logarithmic) densities for automatic sequences along primes and squares. Transactions of the American Mathematical Society 375 (2022), no. 1, 455–499, doi:10.1090/tran/8476; arXiv:2009.14773v2. Theorems 1.2 and 1.4 refer to arXiv version 2, 13 April 2021; journal publication is 2022.
Wouter van Doorn, Partitions with prescribed sum of reciprocals: asymptotic bounds. 2025; arXiv:2502.02200v2. Version 2, 23 July 2025.
Szymon Głąb and Franciszek Prus-Wiśniowski, Achievement sets – current results and open problems. Real Analysis Exchange (2026), doi:10.14321/realanalexch.1766383782; arXiv:2512.17285v1. Advance publication, first available in Project Euclid 8 June 2026. Consulted text remains arXiv:2512.17285v1 (19 December 2025).
Franciszek Prus-Wiśniowski and Jolanta Ptak, Achievable Cantorvals almost without reversed Kakeya conditions. 2024; arXiv:2412.08768v1. Version 1 submitted 11 December 2024. The sparse indices satisfy the overlap inequality, not its strict term-dominating reverse.
Artur Bartoszewicz, Małgorzata Filipczak and Emilia Szymonik, Multigeometric sequences and Cantorvals. Central European Journal of Mathematics 12 (2014), no. 7, 1000–1007, doi:10.2478/s11533-013-0396-4; arXiv:1304.4218v2.
Vivian Kuperberg, Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory 21 (2025), no. 1, 53–74, doi:10.1142/S1793042125500046; arXiv:2301.06095v1. Preprint uploaded 15 January 2023; journal publication is 2025.
Abhishek Jha, The Poisson Tail Conjecture for primes in short intervals. 2026; arXiv:2605.23014v2. Substantially revised version 2, 12 September 2026, 31 pages. Conditional statements must retain their strong Hardy–Littlewood hypotheses.
D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences 1 (2014), article 12, doi:10.1186/s40687-014-0012-7; arXiv:1407.4897v4. Theorem 1.4(i) refers to arXiv version 4 (22 December 2014); the journal numbers it Theorem 4(i). An erratum is recorded at doi:10.1186/s40687-015-0033-x.
Zbigniew Nitecki, Cantorvals and Subsum Sets of Null Sequences. The American Mathematical Monthly 122 (2015), no. 9, 862–870, doi:10.4169/amer.math.monthly.122.9.862; arXiv:1106.3779v2. Consulted preprint: Subsum Sets: Intervals, Cantor Sets, and Cantorvals, version 2 (8 July 2013). Theorem 14 is attributed there to Guthrie–Nymann; its locator is not journal pagination.
Piotr Miska, Franciszek Prus-Wiśniowski and Jolanta Ptak, More on Kakeya Conditions for Achievement Sets. Results in Mathematics 78 (2023), article 113, doi:10.1007/s00025-023-01890-x. Repairs an estimate in the 2021 proof, preserving its uniqueness conclusion, and gives a simpler proof of a weaker theorem without that conclusion.
Jacek Marchwicki and Piotr Miska, On Kakeya Conditions for Achievement Sets. Results in Mathematics 76 (2021), article 181, doi:10.1007/s00025-021-01479-2. Theorem 2.1 is to be read with the proof repair in Miska–Prus-Wiśniowski–Ptak (2023).
Piotr Nowakowski, On a new condition implying that an achievement set is a Cantorval and its applications. 2025; arXiv:2512.17761v1. Version 1, 19 December 2025. Theorem 3.1 requires the Star Procedure of Definition 2 never to break; no application to the present factorial weights is asserted.

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] Paul Erdős, Sur certaines séries à valeur irrationnelle. L’Enseignement Mathématique 4 (1958), 93–100, doi:10.5169/seals-34629.

[ref-2] Paul Erdős and Ronald L. Graham, Old and New Problems and Results in Combinatorial Number Theory. Monographies de L’Enseignement Mathématique, L’Enseignement Mathématique, 1980.

[ref-3] Paul Erdős and Carl Pomerance, On the largest prime factors of $n$ and $n + 1$ . Aequationes Mathematicae 17 (1978), 311–321, doi:10.1007/BF01818569.

[ref-4] Paul Erdős, On the irrationality of certain series: problems and results. New Advances in Transcendence Theory, Cambridge University Press, 1988, pp. 102–109, doi:10.1017/CBO9780511897184.009.

[ref-5] Paul Erdős, Beweis eines Satzes von Tschebyschef. Acta Litterarum ac Scientiarum Szeged 5 (1932), 194–198.

[ref-6] Paul Erdős and Ernst G. Straus, On the irrationality of certain Ahmes series. Journal of the Indian Mathematical Society (N.S.) 27 (1964), 129–133. MR0175848.

[ref-7] Paul Erdős and Ernst G. Straus, On the irrationality of certain series. Pacific Journal of Mathematics 55 (1974), no. 1, 85–92, doi:10.2140/pjm.1974.55.85.

[ref-8] Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, Long gaps between primes. Journal of the American Mathematical Society 31 (2018), 65–105, doi:10.1090/jams/876; arXiv:1412.5029v3.

[ref-9] John A. Fridy, Generalized bases for the real numbers. The Fibonacci Quarterly 4 (1966), no. 3, 193–201.

[ref-10] A. Ya. Khinchin, Continued Fractions. University of Chicago Press, 1964.

[ref-11] 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-12] Vivian Kuperberg, Sums of singular series with large sets and the tail of the distribution of primes. The Quarterly Journal of Mathematics 74 (2023), no. 4, 1457–1479, doi:10.1093/qmath/haad030; arXiv:2210.09775v2. Statement numbers refer to arXiv version 2, 15 June 2023.

[ref-13] Johan Land, A conditional proof of the irrationality of $\sum_{n \geq 1} p_{n} 2^{- n}$ under a uniform Hardy–Littlewood prime-tuples conjecture. Research draft, 5 September 2026. Formalisation material accompanies the draft.

[ref-14] Leonardo de Moura and Sebastian Ullrich, The Lean 4 theorem prover and programming language. Automated Deduction – CADE 28, Lecture Notes in Computer Science, Springer, 2021, pp. 625–635, doi:10.1007/978-3-030-79876-5_37.

[ref-15] The mathlib Community, The Lean mathematical library. Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs, ACM, 2020, pp. 367–381, doi:10.1145/3372885.3373824. Describes a Lean 3-era snapshot; the repository lock identifies the library used by the present Lean 4 sources.

[ref-16] James Maynard, Small gaps between primes. Annals of Mathematics 181 (2015), 383–413, doi:10.4007/annals.2015.181.1.7.

[ref-17] Hugh L. Montgomery and Robert C. Vaughan, Multiplicative Number Theory I: Classical Theory. Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2007, doi:10.1017/CBO9780511618314.

[ref-18] Jan-Christoph Schlage-Puchta, The irrationality of some number theoretical series. Acta Arithmetica 126 (2007), no. 4, 295–303, doi:10.4064/aa126-4-1; arXiv:1105.1451v1. Published in 2007; arXiv upload is from 2011. Lemma and preprint page locators refer to arXiv version 1.

[ref-19] Ian Short, Ford Circles, Continued Fractions, and Rational Approximation. The American Mathematical Monthly 118 (2011), no. 2, 130–135, doi:10.4169/amer.math.monthly.118.02.130; arXiv:0912.1997v1. Preprint title: Ford circles, continued fractions, and best approximation of the second kind. Preprint theorem locators refer to arXiv version 1.

[ref-20] 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-21] Yitang Zhang, Bounded gaps between primes. Annals of Mathematics 179 (2014), 1121–1174, doi:10.4007/annals.2014.179.3.7.

[ref-22] Thomas F. Bloom, Erdős Problem #251. 2026. Accessed 6 September 2026.

[ref-23] Erdős Problems contributors, Erdős Problem #251 discussion thread. 2026. Accessed 6 September 2026: Tao comment of 7 October 2025 and Land comments of 6 September 2026.

[ref-24] ChatGPT 5.4 Pro (orchestrated by Vjeko Kovač), On the Erdős problem #251. Unpublished note, Department of Mathematics, University of Zagreb, 2026. Accessed 6 September 2026.

[ref-25] The Formal Conjectures Authors, FormalConjectures.ErdosProblems.251. 2025. Pinned statement source, not a proof of Problem 251; accessed 28 July 2026.

[ref-26] Stefan Ringer, Local gap statistics, telescoping, and normality: a local-pattern approach to Erdős problem 251. Preprint, 11 September 2026. Mutable main-branch TeX consulted 16 September 2026; the cited conditional statements were rechecked 18 September 2026. Formalisation material accompanies the draft.

[ref-27] Tonći Crmarić and Vjekoslav Kovač, On the irrationality of certain super-polynomially decaying series. Colloquium Mathematicum 179 (2025), 55–68, doi:10.4064/cm9628-5-2025; arXiv:2504.18712v1. Lemma 4 is cited using arXiv version 1.

[ref-28] Boris Adamczewski, Michael Drmota and Clemens Müllner, (Logarithmic) densities for automatic sequences along primes and squares. Transactions of the American Mathematical Society 375 (2022), no. 1, 455–499, doi:10.1090/tran/8476; arXiv:2009.14773v2. Theorems 1.2 and 1.4 refer to arXiv version 2, 13 April 2021; journal publication is 2022.

[ref-29] Wouter van Doorn, Partitions with prescribed sum of reciprocals: asymptotic bounds. 2025; arXiv:2502.02200v2. Version 2, 23 July 2025.

[ref-30] Szymon Głąb and Franciszek Prus-Wiśniowski, Achievement sets – current results and open problems. Real Analysis Exchange (2026), doi:10.14321/realanalexch.1766383782; arXiv:2512.17285v1. Advance publication, first available in Project Euclid 8 June 2026. Consulted text remains arXiv:2512.17285v1 (19 December 2025).

[ref-31] Franciszek Prus-Wiśniowski and Jolanta Ptak, Achievable Cantorvals almost without reversed Kakeya conditions. 2024; arXiv:2412.08768v1. Version 1 submitted 11 December 2024. The sparse indices satisfy the overlap inequality, not its strict term-dominating reverse.

[ref-32] Artur Bartoszewicz, Małgorzata Filipczak and Emilia Szymonik, Multigeometric sequences and Cantorvals. Central European Journal of Mathematics 12 (2014), no. 7, 1000–1007, doi:10.2478/s11533-013-0396-4; arXiv:1304.4218v2.

[ref-33] Vivian Kuperberg, Sums of singular series along arithmetic progressions and with smooth weights. International Journal of Number Theory 21 (2025), no. 1, 53–74, doi:10.1142/S1793042125500046; arXiv:2301.06095v1. Preprint uploaded 15 January 2023; journal publication is 2025.

[ref-34] Abhishek Jha, The Poisson Tail Conjecture for primes in short intervals. 2026; arXiv:2605.23014v2. Substantially revised version 2, 12 September 2026, 31 pages. Conditional statements must retain their strong Hardy–Littlewood hypotheses.

[ref-35] D. H. J. Polymath, Variants of the Selberg sieve, and bounded intervals containing many primes. Research in the Mathematical Sciences 1 (2014), article 12, doi:10.1186/s40687-014-0012-7; arXiv:1407.4897v4. Theorem 1.4(i) refers to arXiv version 4 (22 December 2014); the journal numbers it Theorem 4(i). An erratum is recorded at doi:10.1186/s40687-015-0033-x.

[ref-36] Zbigniew Nitecki, Cantorvals and Subsum Sets of Null Sequences. The American Mathematical Monthly 122 (2015), no. 9, 862–870, doi:10.4169/amer.math.monthly.122.9.862; arXiv:1106.3779v2. Consulted preprint: Subsum Sets: Intervals, Cantor Sets, and Cantorvals, version 2 (8 July 2013). Theorem 14 is attributed there to Guthrie–Nymann; its locator is not journal pagination.

[ref-37] Piotr Miska, Franciszek Prus-Wiśniowski and Jolanta Ptak, More on Kakeya Conditions for Achievement Sets. Results in Mathematics 78 (2023), article 113, doi:10.1007/s00025-023-01890-x. Repairs an estimate in the 2021 proof, preserving its uniqueness conclusion, and gives a simpler proof of a weaker theorem without that conclusion.

[ref-38] Jacek Marchwicki and Piotr Miska, On Kakeya Conditions for Achievement Sets. Results in Mathematics 76 (2021), article 181, doi:10.1007/s00025-021-01479-2. Theorem 2.1 is to be read with the proof repair in Miska–Prus-Wiśniowski–Ptak (2023).

[ref-39] Piotr Nowakowski, On a new condition implying that an achievement set is a Cantorval and its applications. 2025; arXiv:2512.17761v1. Version 1, 19 December 2025. Theorem 3.1 requires the Star Procedure of Definition 2 never to break; no application to the present factorial weights is asserted.