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.

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In this paper

Summation by parts, with the endpoint retained

We return to the actual prime gaps $g_{n} = p_{n + 1} - p_{n}$ , with $p_{0} = 2$ and $g_{0} = 1$ . We write

Π = \sum_{n \geq 0} p_{n} 2^{- n - 1}, S = \sum_{n \geq 0} g_{n} 2^{- n - 1} .

Here $S$ denotes the gap sum. The set of indices used in Section 2 will not be needed below. The short paper calls this same sum $G$ . The two calculations below have different jobs: summation by parts gives $Π = 2 + S$ , and deleting the first gap gives the initial tail $T_{0} = 2 S - 1$ .

We begin with a finite identity, so no convergence hypothesis is needed. Its endpoint identifies the estimate required before passing to the infinite series. The linked definitions are the dyadic partial sum and the weighted sum of consecutive differences. The formula below writes both sums explicitly.

Proposition 4.1 (finite summation by parts). For every rational sequence $P$ and every $n \geq 0$ ,

\sum_{i = 0}^{n} \frac{P (i)}{2^{i + 1}} = P (0) + \sum_{i = 0}^{n - 1} \frac{P (i + 1) - P (i)}{2^{i + 1}} - \frac{P (n)}{2^{n + 1}} .

Proof. At $n = 0$ both sides equal $P (0) / 2$ . When $n$ is replaced by $n + 1$ , the change on the right is

\frac{P (n + 1) - P (n)}{2^{n + 1}} - \frac{P (n + 1)}{2^{n + 2}} + \frac{P (n)}{2^{n + 1}} = \frac{P (n + 1)}{2^{n + 2}} .

This is the next term on the left, which proves the identity by induction. ◻

Specialising to $P (i) = p_{i}$ , whose first value is $p_{0} = 2$ , and writing $g_{i} = p_{i + 1} - p_{i}$ for the zero-based gaps, gives the reformulation.

Theorem 4.2 (prime-gap reformulation). Let $p_{0} = 2, p_{1} = 3, \dots$ be the primes in increasing order and $g_{i} = p_{i + 1} - p_{i}$ . For every $n \geq 0$ ,

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

The leading $2$ is the first prime, not a normalising constant. At $n = 2$ , for instance, the left side is $2 / 2 + 3 / 4 + 5 / 8 = 19 / 8$ and the right side is $2 + (1 / 2 + 2 / 4) - 5 / 8 = 19 / 8$ .

The infinite identity and the irrationality equivalence

To pass to the infinite series, let

u_{n} = \frac{p_{n}}{2^{n + 1}}, v_{n} = \frac{g_{n}}{2^{n + 1}} .

Thus $\sum_{n} u_{n} = Π$ . The termwise identity $v_{n} = 2 u_{n + 1} - u_{n}$ is the dyadic discrete derivative. It expresses each gap term as an integer combination of two consecutive prime terms, so once $(u_{n})$ is summable the gap series can be summed after proving absolute convergence, which justifies shifting and subtracting the prime series.

Theorem 4.3 (infinite prime-gap identity). Both series converge and

\sum_{n \geq 0} \frac{p_{n}}{2^{n + 1}} = 2 + \sum_{n \geq 0} \frac{g_{n}}{2^{n + 1}} .

Proof. The bound $p_{n} \leq 1250 (n + 1)^{4}$ of Appendix A makes $(u_{n})$ summable, and $0 < g_{n} \leq p_{n + 1}$ then makes $(v_{n})$ summable. The shifted sequence $(u_{n + 1})$ is summable, so summing $v_{n} = 2 u_{n + 1} - u_{n}$ and using $u_{0} = 1$ gives

\sum_{n \geq 0} v_{n} = 2 (\sum_{n \geq 0} u_{n} - u_{0}) - \sum_{n \geq 0} u_{n} = \sum_{n \geq 0} u_{n} - 2 . ◻

◻

The prime number theorem is needed neither for this convergence nor for the identity. Its asymptotic $p_{n} \sim n \log n$ is used in Corollary 8.4 and in the argument counting repeated tail values [17].

Corollary 4.4 (exact irrationality reformulation). $Π$ is irrational if and only if $S = \sum_{n \geq 0} g_{n} 2^{- (n + 1)}$ is irrational. The corresponding zero-based series with denominator $2^{n}$ equals $4 + 2 S$ and has the same irrationality status.

Proof. The identity just proved gives $Π = 2 + S$ . Adding a rational constant or multiplying by a nonzero rational number preserves rationality and irrationality. ◻

Concretely, $Π = 3.674643966 \dots$ is irrational if and only if $S = 1.674643966 \dots$ is. Neither irrationality statement is established here.

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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