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

PDFOpen the PDF, 47 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

Algebraic nonconcentration survives the rationalising perturbation

Schlage-Puchta’s lemma states that $F (g_{n}, \dots, g_{n + k}) \neq 0$ for almost all $n$ , for every fixed nonzero $F \in Z [x_{0}, \dots, x_{k}]$ [18]. We say that an integer sequence has fixed-block nonconcentration if it satisfies this conclusion for every $k \geq 0$ and every such $F$ . A bounded integer perturbation preserves the property: there are only finitely many translated polynomials to consider for each fixed block. No randomness or independence assumption is required.

This nonconcentration condition excludes every bounded integer sequence: if its values lie in a finite set $E$ , the nonzero polynomial $\prod_{c \in E} (x_{0} - c)$ vanishes at every index. It also excludes a sequence satisfying a fixed nonzero polynomial relation on every short block, such as $a_{n} = n$ , for which $x_{1} - x_{0} - 1$ vanishes. The density-zero exceptional set depends on the fixed polynomial. There cannot be one such set for all polynomials: $x_{0} - a_{n}$ vanishes on the block beginning at $n$ , so a common exceptional set would contain every index. No uniform estimate for growing families of polynomials or block lengths is asserted.

In the next proposition, $N, H$ are nonnegative integers, $h$ is an integer, and all counts are over integer indices and integer triples.

Proposition 8.2 (finite counting for shifted gap differences). Write $p_{n}$ for the primes indexed from $p_{0} = 2$ and $g_{n} = p_{n + 1} - p_{n}$ . For $h \geq 2$ and $r \in Z$ , let $M_{h, r} (N)$ count $n < N$ with $g_{n + h} - g_{n} = r$ . Let $Q_{N, H, r}$ count triples $(x, d, s)$ with $x < p_{N}$ , $0 < d < s \leq H$ , $d + r > 0$ , and all four integers $x, x + d, x + s, x + s + d + r$ prime. Then, for every $N, H \geq 0$ ,

(H + 1) M_{h, r} (N) \leq (h + 1) p_{N + h + 1} + (H + 1) Q_{N, H, r} .

Proof. For a counted index $n$ , put $x = p_{n}$ , $d = g_{n}$ and $s = p_{n + h} - p_{n}$ . If the total span $p_{n + h + 1} - p_{n}$ is at most $H$ , then $0 < d < s \leq H$ , $d + r = g_{n + h} > 0$ , and the four primes are precisely $x, x + d, x + s, x + s + d + r$ . The count $Q_{N, H, r}$ need not require these primes to be consecutive: it is an upper bound, and $x = p_{n}$ determines $n$ , making the map injective. For the remaining indices the integer span is at least $H + 1$ . Each gap appears in at most $h + 1$ of the spans, so their total is at most $(h + 1) \sum_{i < N + h} g_{i} < (h + 1) p_{N + h + 1}$ . Multiply the resulting bound on the exceptional count by $H + 1$ and add the small-span contribution. ◻

Increasing $H$ reduces the bound for long spans but enlarges the prime-pattern count. To deduce zero density from this finite inequality, for each $ε > 0$ and all large $N$ one needs a single choice of $H$ making the entire right side less than $ε N (H + 1)$ . For $h = 1$ , the same definitions give $s = d$ , so $x + d = x + s$ . The strict condition $d < s$ in the four-prime count fails; that case requires a three-prime count instead. Zero density itself is already known for every fixed $h \geq 1$ and $r \in Z$ : apply Schlage-Puchta’s lemma above to the nonzero polynomial $x_{h} - x_{0} - r$ . Thus the finite reduction should not be read as leaving that upper bound open. Neither upper bound supplies the late occurrences or weighted-tail separation required for irrationality.

For the perturbation claim, first suppose that $a$ has fixed-block nonconcentration and every correction is either $0$ or a fixed integer $M \geq 1$ . For a fixed $r \in Z$ , an equality $b_{n + 1} - b_{n} = r$ then requires $a_{n + 1} - a_{n} \in {r - M, r, r + M}$ . Each of these three equalities holds on a set of density zero. The general argument uses the same finite-union step for translated polynomials.

Theorem 8.3 (nonconcentration is perturbation-stable). Let $a : N \to Z$ have fixed-block nonconcentration, let $E \subset Z$ be finite, and let $b_{n} = a_{n} + e_{n}$ with $e_{n} \in E$ for every $n$ . Then $b$ has fixed-block nonconcentration.

Proof. Fix $k \geq 0$ and a nonzero $F \in Z [x_{0}, \dots, x_{k}]$ . For a tuple $e = (e^{(0)}, \dots, e^{(k)}) \in E^{k + 1}$ put $F_{e} (x_{0}, \dots, x_{k}) = F (x_{0} + e^{(0)}, \dots, x_{k} + e^{(k)})$ . The substitution $x_{i} \mapsto x_{i} + e^{(i)}$ is a ring automorphism of $Z [x_{0}, \dots, x_{k}]$ with inverse $x_{i} \mapsto x_{i} - e^{(i)}$ , so $F_{e} \neq 0$ . If $F (b_{n}, \dots, b_{n + k}) = 0$ then $F_{e} (a_{n}, \dots, a_{n + k}) = 0$ for the tuple $e = (e_{n}, \dots, e_{n + k})$ , whence

{n : F (b_{n}, \dots, b_{n + k}) = 0} \subseteq ⋃_{e \in E^{k + 1}} {n : F_{e} (a_{n}, \dots, a_{n + k}) = 0} .

Each set on the right has density zero by hypothesis, and there are only finitely many such sets because $E^{k + 1}$ is finite. Their union therefore has density zero. ◻

Finiteness of the correction set is essential to this argument; no independence between the corrections and the sequence is needed. Without a restriction on the corrections, the choice $e_{n} = - a_{n}$ would give $b_{n} = 0$ , which fails nonconcentration even for the polynomial $x_{0}$ .

Corollary 8.4 (nonconcentration does not force irrationality). Fix $M \geq 1$ and $K \geq 0$ , and let $b$ be the perturbed sequence supplied by Theorem 8.1 at the actual prime gaps. Then $\sum_{n \geq 0} b_{n} 2^{- (n + 1)}$ is rational, $b_{n} = g_{n}$ for $n < K$ , $b_{n} - g_{n} \in {0, M}$ and $b_{n} \equiv g_{n} (\mod M)$ for every $n$ , $b$ has fixed-block nonconcentration, and the cumulative sequence $P_{n} = 2 + \sum_{i < n} b_{i}$ satisfies $p_{n} \leq P_{n} \leq p_{n} + M n$ and hence $P_{n} \sim n \log n$ .

Proof. Theorem 8.1 gives the rational sum, prefix and congruence assertions. Summing $0 \leq b_{i} - g_{i} \leq M$ and using $\sum_{i < n} g_{i} = p_{n} - 2$ gives $p_{n} \leq P_{n} \leq p_{n} + M n$ . Now $p_{n} \sim n \log n$ [17] implies $P_{n} \sim n \log n$ . The perturbation takes values in the fixed two-element set $E = {0, M}$ , so Theorem 8.3 applies to the actual gaps, which have fixed-block nonconcentration by Schlage-Puchta’s lemma [18]. ◻

All the corollary’s properties hold for one rational-sum sequence; the positions $P_{n}$ need not be prime. Taking $M$ even also preserves eventual evenness. In particular $M = 2 q$ preserves parity and modulus $q$ , with $b_{n} \leq g_{n} + 2 q$ . Nonconcentration remains a fixed-polynomial, fixed-block assertion, not a uniform estimate for growing families.

Proposition 8.5 (nonconcentration under sparse changes). Let $a, b : N \to Z$ agree off a set $S$ of ordinary density zero. If $a$ has fixed-block nonconcentration, then so does $b$ . No boundedness assumption on $a - b$ is needed.

Proof. Fix $k$ and a nonzero $F \in Z [x_{0}, \dots, x_{k}]$ . The two blocks agree unless $n + i \in S$ for some $0 \leq i \leq k$ . Consequently

\begin{aligned} | {n < N : F (b_{n}, \dots, b_{n + k}) = 0} | \\ \leq | {n < N : F (a_{n}, \dots, a_{n + k}) = 0} | + \sum_{i = 0}^{k} | S \cap [i, N + i) | . \end{aligned}

The first count is $o (N)$ by nonconcentration for $a$ ; the remaining finitely many counts are $o (N)$ because $S$ has density zero. The count concerns the event $F = 0$ , not the sizes of $F$ or the corrections. The sequence $b$ is already fixed, so this holds for every fixed $F, k$ , without a uniform estimate for growing polynomial families or block lengths. ◻

Corollary 8.6 (a rational sum with the stated prime-gap statistics). For every prescribed finite prime-gap prefix and $0 < ε \leq 1$ , there is an altered sequence $b = g + e$ , with $P_{n} = 2 + \sum_{i < n} b_{i}$ , for which one can simultaneously impose a rational dyadic value, nonnegative integer corrections eventually at most $(\log (n + 3))^{ε}$ , all fixed eventual coefficient and cumulative congruences, fixed-block polynomial nonconcentration, and vanishing total variation distance between unnormalised block distributions for lengths $o (\log \log X)$ . The cumulative positions satisfy

0 \leq P_{n} - p_{n} = O_{ε} (\frac{n (\log (n + 3))^{ε}}{\log \log n}), P_{n} \sim n \log n .

Proof. Apply Theorem 2.1 to the actual gaps, choose a rational target in its interval, and fix the resulting $b = g + e$ . Proposition 8.5 then transfers Schlage-Puchta’s nonconcentration [18] to this one sequence, for every fixed polynomial and block length. To obtain the cumulative support bound, split $[\sqrt{n}, n)$ into dyadic intervals, use $\log \log x ≍ \log \log n$ there, and bound the first $\sqrt{n}$ indices trivially. Thus $| S \cap [0, n) | = O (n / \log \log n)$ . The pointwise correction bound gives the displayed estimate, including a fixed constant for the exceptional prefix. Since $(\log n)^{ε - 1} / \log \log n \to 0$ for $0 < ε \leq 1$ , the prime number theorem [17] gives the last conclusion. The position series is rational as well: reversing the order of summation of nonnegative terms gives

\sum_{n \geq 0} \frac{P_{n}}{2^{n + 1}} = 2 + \sum_{i \geq 0} b_{i} \sum_{n > i} 2^{- n - 1} = 2 + \sum_{i \geq 0} \frac{b_{i}}{2^{i + 1}} \in Q .

The finite value on the right also proves convergence on the left. ◻

The case $ε > 1$ needs no new construction: use exponent $1$ . Its corrections satisfy the larger allowance, while the stronger error $O (n \log (n + 3) / \log \log n) = o (n \log n)$ still preserves cumulative growth. Thus every positive exponent is available. The construction supplies the support and correction estimates; Schlage-Puchta’s lemma supplies nonconcentration of the original gaps, and the prime number theorem supplies their cumulative asymptotic.

For each fixed integer $y \geq 2$ , take modulus $y!$ . Once the congruence holds and $p_{n} > y$ , $gcd (P_{n}, y!) = 1$ excludes every prime divisor at most $y$ . This is stronger than eventual oddness. The cutoff depends on $y$ ; primality would require this exclusion up to $\sqrt{P_{n}}$ at the same index. Even prime-valued positions could omit intervening primes.

The bounded construction transfers nonconcentration by finitely many polynomial translates, not by preserving block frequencies. The sparse construction transfers block laws by leaving most blocks unchanged, with absolute error tending to zero for $m = o (\log \log X)$ . Neither argument preserves relative rare-event frequencies or the quantitative prime-pattern hypotheses in Section 3.

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.

Prefer the manuscript?

PDFOpen the PDF, 47 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:db2a5b20038e07a9. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.