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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In this paper

Introduction

The primes $p_{0} = 2, p_{1} = 3, \dots$ define the dyadic series

Π = \sum_{n \geq 0} \frac{p_{n}}{2^{n + 1}} = 2 / 2 + 3 / 4 + 5 / 8 + 7 / 16 + \dots .

Its irrationality is Erdős Problem #251 [22], recorded in [1], [2] and [4]. Writing $g_{n} = p_{n + 1} - p_{n}$ , summation by parts gives $Π = 2 + \sum_{n} g_{n} 2^{- (n + 1)}$ . We use this identity to compare what can be deduced from properties of the coefficients with what would be needed from the actual prime-gap tails.

Theorem 2.1 shows that a substantial collection of coefficient properties does not determine rationality among comparison sequences. Starting with any convergent nonnegative integer dyadic series, it attains an interval of sums using corrections on one sparse set. That set and every eventual coefficient and partial-sum congruence cutoff are fixed before the target. The family therefore contains both rational and irrational sums, not merely a separately chosen example for each preserved property. The corrections can respect any prescribed bound tending to infinity.

At each stage a fixed ordinary correction total protects the congruence calculations at later stages. Redistributing that total between adjacent indices changes its dyadic contribution. Growing digit ranges compensate for the dyadic decay across the increasingly separated stages. A telescoping estimate then ensures that the remaining range spans every gap between adjacent present choices. With a polylogarithmic correction bound, the empirical block laws remain asymptotically equal for lengths $o (\log \log X)$ sampled in $[X, 2 X)$ . For prime gaps, polynomial nonconcentration and cumulative growth $n \log n$ also survive. The missing requirement is that the cumulative positions be consecutive primes; neither residues nor these statistics supply it.

The actual-prime arguments, beginning in Section 4, are independent of the construction. Summation by parts and a vanishing boundary condition identify the complete tails. Integral shifts then give exact rationality criteria; two consecutive small differences with unequal gaps give a sufficient condition for irrationality. Section 6 certifies individual differences, while Section 7 excludes finite ranges of rational denominators. Neither finite calculation supplies arbitrarily late witnesses. Section 8 tests weaker coefficient conditions by counterexamples; the appendices retain the integer calculations, further criteria and source concordance.

For reference, the first primes and gaps are

\begin{array}{ccccccccccc} i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ p_{i} & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 \\ g_{i} & 1 & 2 & 2 & 4 & 2 & 4 & 2 & 4 & 6 & 2 \end{array}

Thus $g_{0} = 1$ and all later gaps are even. This parity restriction will be used to characterise two consecutive tail differences of absolute value less than one. The alternative normalisation with denominator $2^{i}$ satisfies

\sum_{i = 0}^{n - 1} \frac{p_{i}}{2^{i}} = 2 \sum_{i = 0}^{n - 1} \frac{p_{i}}{2^{i + 1}},

so it has the same rationality status. The formal factor-of-two identity uses these same finite sums.

Notation.

We write $N = {0, 1, \dots}$ , use $φ$ for Euler’s totient function, and let $den q$ be the positive denominator of $q \in Q$ in lowest terms. A real number is integral if it belongs to $Z$ , and $dist (x, Z)$ is its distance to the nearest integer. A sequence is eventually periodic with period $h \geq 1$ if $a_{n + h} = a_{n}$ for every sufficiently large $n$ . A set of indices has density zero if its counting function is $o (x)$ , and a property holds for almost all indices if its exceptional set has density zero. A property may hold at arbitrarily large indices even when those indices form a set of density zero. An empty sum is zero.

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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