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.

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Remaining prime-gap estimates

For the actual tails, Theorem 5.3 makes the first question equivalent to irrationality. The second requires two small differences and a gap mismatch at the same index. A finite certificate verifies one instance; both questions require witnesses arbitrarily late for every positive shift.

Problem 10.1 (nonintegral tail differences of each positive length). For every $h \geq 1$ and every $N_{0}$ , prove that some $N \geq N_{0}$ satisfies

\begin{matrix} (9) & \sum_{j \geq 1} \frac{g_{N + h + j} - g_{N + j}}{2^{j}} \notin Z . \end{matrix}

Problem 10.2 (two small tail differences at arbitrarily large indices). For every $h \geq 1$ and every $N_{0}$ , prove that some $N \geq N_{0}$ satisfies

\begin{matrix} (10) & | σ_{h} (N) | < 1, | σ_{h} (N + 1) | < 1, g_{N + h + 1} \neq g_{N + 1} . \end{matrix}

Problem 10.1, like the congruent-index criterion of Theorem 5.5, is exact; Problem 10.2 is only sufficient. The counterexamples in Section 8 show why the retained coefficient data do not supply the missing tail information. They are not counterexamples about actual primes.

The successful indices have density zero by Theorem 8.7, so the required existence theorem must detect a sparse set. A finite prefix is compatible with a rational comparison series. Supplemented by a proved remainder bound, however, it can certify one actual pair, as in Proposition 6.6; it does not establish occurrence beyond every cutoff.

Proposition 6.5 gives sufficient integer inequalities ensuring $| σ_{h} (N) | < 1$ at a chosen truncation length $L$ . Failure of such a test need not mean that the complete tail lies outside the required interval: its error interval may still cross an endpoint. For fixed $h$ and $ε > 0$ , the choice $L = ⌈ (4 + ε) \log_{2} (N + 2) ⌉$ gives $E_{h, N, L} = O_{h, ε} (N^{- ε})$ . Indeed, $L = O_{ε} (\log N)$ , both arguments of the quartic $P$ are $O_{h, ε} (N)$ , and $2^{- L} \leq (N + 2)^{- 4 - ε}$ . Requiring a fixed positive margin from both endpoints is stronger than the strict inequalities in (10). At this depth, the test bounds the signed integer $2^{L} F_{h, N, L}$ , not just its residue modulo $2^{L}$ , and also requires $g_{N + h + 1} - g_{N + 1} = \pm 2$ . For fixed $h$ and $ε$ , checking one certificate uses $O_{ε} (\log N)$ gap values and the explicit remainder bound. This counts data, not bit operations. The unproved input is the existence of certified witnesses beyond every cutoff for each fixed $h$ ; constants uniform in $h$ are not required.

The following published results control gap sizes and clusters, not the signed weighted sums required by this criterion. Zhang’s bounded-gap theorem [21] produces infinitely many bounded consecutive-prime gaps. Maynard bounds $\underset{n}{lim inf} (p_{n + m} - p_{n})$ for every fixed $m$ , giving bounded clusters of every fixed size [16], with the explicit unconditional bound $\underset{n}{lim inf} (p_{n + 1} - p_{n}) \leq 600$ [16]. Polymath subsequently improved this to $246$ [35]; neither numerical bound controls the signed weighted tails required here. The large-gap theorem of Ford, Green, Konyagin, Maynard and Tao gives an effective lower bound for the largest single consecutive-prime gap below $X$ [8]. A theorem giving bounded clusters at each fixed cluster size does not by itself supply a weighted condition on blocks whose length grows with the basepoint, and none of these results is used as a proof input here.

Conditionally the picture is different. Tao’s comment names uniform quantitative prime-tuples control of about $\log \log n$ consecutive gaps as a possible approach [23], and Land’s draft carries that out under Kuperberg’s uniform prime-tuples conjecture [13]. The counterexamples in Section 8 show that the listed size, congruence and block-statistical properties alone do not force irrationality. These constructions are not shown to enumerate the consecutive primes, so they do not rule out arguments that combine those properties with further information about primes.

Funding and competing interests.

This work received no external funding. The author declares no competing interests.

Acknowledgements.

The problem numbering follows the Erdős Problems catalogue maintained by Thomas Bloom [22]. The logarithmic-scale construction and the transfer of nonconcentration come from earlier working materials of this project, which were not an independent mathematical review.

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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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