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

Will Cook

Long paper

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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Polynomial coefficients and telescoping sums

We can construct further rational sums by starting with a polynomial $U_{n}$ and defining $c_{n + 1} = 2 U_{n} - U_{n + 1}$ . Polynomial growth gives the vanishing dyadic boundary condition, so the recurrence identifies $U_{n}$ as the complete tail. A quadratic choice already makes the coefficients positive and increasing.

Proposition 8.9 (a quadratic sequence with integral tails). Put $c_{n} = 2 (n^{2} + 4 n + 2)$ and $U_{n} = 2 (n + 4)^{2}$ . Then $c_{n}$ is positive, even and strictly increasing, $U_{n + 1} = 2 U_{n} - c_{n + 1}$ , every shift $U_{N + h} - U_{N}$ is integral, $c_{n + 1} - c_{n} = 4 n + 10$ is never $\pm 2$ , and

\sum_{j \geq 1} \frac{c_{j}}{2^{j}} = 32 .

Proof. Direct expansion gives the recurrence, and the finite telescope is $\sum_{j = 1}^{n} c_{j} 2^{- j} = 32 - 2 (n + 4)^{2} 2^{- n}$ , whose last term tends to zero. The remaining assertions follow from the integer values of $U_{n}$ and from $c_{n + 1} - c_{n} = 4 n + 10 \geq 10$ . ◻

The series starts at $j = 1$ and equals the initial rescaled tail $U_{0} = 32$ . Under the opening convention its sum is $\sum_{n \geq 0} c_{n} 2^{- (n + 1)} = (c_{0} + 32) / 2 = 18$ . Positivity, parity, strict growth, unboundedness and nonperiodicity are therefore jointly compatible with rationality, and the adjacent-gap condition of Theorem 6.4 fails at every index. The telescoping mechanism is the one used in the note of ChatGPT 5.4 Pro (orchestrated by Vjeko Kovač) against the variable-denominator expectation of Erdős [24]; the sequence is not a result about Problem #251.

Rationality also fails to force the coefficients to repeat. Let $K : N \to Q$ be arbitrary and put $κ_{n} = 2 K_{n} - K_{n + 1}$ : the coefficient of the telescoping series.

Proposition 8.10 (exact telescoping). For every $n \geq 0$ , $\sum_{i = 0}^{n - 1} κ_{i} 2^{- (i + 1)} = K_{0} - K_{n} 2^{- n}$ .

Proof. The sum is empty when $n = 0$ . Passing from $n$ to $n + 1$ adds $(2 K_{n} - K_{n + 1}) 2^{- (n + 1)} = K_{n} 2^{- n} - K_{n + 1} 2^{- (n + 1)}$ , which replaces the terminal term by the required one. ◻

The identity $κ_{n} = 2 K_{n} - K_{n + 1}$ has the algebraic form used in the rationality criterion of Erdős and Straus when $a_{n} = 2$ . That criterion has additional hypotheses: for integers $b_{n}$ and positive integers $a_{n}$ with $a_{n} > 1$ for all large $n$ and $| b_{n} | / (a_{n - 1} a_{n}) \to 0$ , the series $\sum_{n} b_{n} / (a_{1} \dots a_{n})$ is rational exactly when some positive integer $B$ and integers $c_{n}$ satisfy $B b_{n} = c_{n} a_{n} - c_{n + 1}$ and $| c_{n + 1} | < a_{n} / 2$ for all large $n$ [7]. With $a_{n} = 2$ , the smallness hypothesis becomes $| b_{n} | / 4 \to 0$ . Since the $b_{n}$ are integers, this forces $b_{n} = 0$ eventually. Thus the criterion in this specialisation does not apply even to a bounded integer sequence that is nonzero infinitely often, let alone to the positive prime gaps. Proposition 8.10, by contrast, imposes no growth condition on $K$ . The infinite series converges precisely when $2^{- n} K_{n}$ has a finite limit, and then its sum is $K_{0} - lim_{n \to \infty} 2^{- n} K_{n}$ . The sum equals $K_{0}$ precisely when that limit is zero; polynomial growth suffices. For $K_{n} = 2^{n}$ , every $κ_{n}$ is zero, so the series converges to $0$ , not to $K_{0} = 1$ .

Taking $K_{0} = \frac{5}{2}$ and $K_{n} = 2 n + 2$ for $n \geq 1$ gives $κ_{0} = 1$ and $κ_{n} = 2 n$ , so the coefficients $1, 2, 4, 6, 8, \dots$ are positive, even after the first term, unbounded and not eventually periodic, while their dyadic sum is $\frac{5}{2}$ . Rationality alone therefore cannot imply eventual periodicity even for a positive, parity-correct integer coefficient sequence, and rationality cannot be contradicted merely by Proposition 6.3.

The arithmetic-progression formulations of nonintegrality do not create new prime-distribution information. Their full definitions and exact hypotheses are preserved in Appendix E.

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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