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

Source index for the short paper

The links below identify further formal definitions and statements used in the short paper, at its original revision 99f4bf47422a and line numbers. This local source pin applies only to the index below; Appendix C retains the long record’s separate pin. Each link is labelled by the mathematics it states.

Finite series. The first prime gap is one; The second prime gap is two; Finite dyadic sums; Finite dyadic sums of consecutive differences; Summation by parts for a rational sequence; The two prime-series normalisations differ by a factor of two; Summation by parts for the primes; Terms of the prime series; Terms of the prime-gap series; The identity relating prime and gap terms.

Convergent series. Convergence of the gap series from convergence of the prime series; The infinite prime-to-gap identity under summability; Prime terms with denominator powers starting at zero; The factor-of-two identity for individual terms; Equivalent irrationality of the prime and gap series; The infinite identity in the second normalisation; Equivalent irrationality in the second normalisation; An elementary polynomial upper bound for the primes; Convergence of the prime series; The prime-to-gap identity with convergence proved.

Recurrences. The recurrence satisfied by rescaled dyadic tails; The difference between two terms of a recurrence; The weighted integer sum accumulated by iteration; Iteration of the recurrence; A shift as a multiple of a tail minus an integer; The recurrence for consecutive tail differences; Subtracting an integer preserves integrality; The criterion for a shift to be integral.

Denominators and integral shifts. The denominator divisibility criterion for an integral shift; Euler’s congruence for an odd denominator; An integral shift of totient length; An integral shift remains integral at the next index; An integral shift remains integral at every later index; The denominator after one recurrence step; An odd denominator is unchanged by a step; An even denominator is halved by a step; Removing the power of two from a rational denominator; An integral multiple after the power of two is removed; Eventual integral shifts from the denominator factorisation; Eventual integral shifts under rationality.

Rationality criteria. Rational representation of a rescaled prime-gap tail; The recurrence for that rational representation; Existence of the rational representation under rationality; Rationality is equivalent to one integral positive-length shift; Rationality is equivalent to eventual integrality of a fixed shift; Irrationality is equivalent to nonintegrality of every positive-length shift; Irrationality is equivalent to arbitrarily late nonintegral shifts of each length.

Small differences and counterexamples. Two small differences with unequal coefficients cannot both be integral; Arbitrarily late small unequal pairs exclude eventual integrality; The small-pair implication for prime gaps; Prime gaps exceed every prescribed bound; Prime gaps are not eventually periodic; Coefficients obtained by telescoping; The finite telescoping identity; The telescoping example with coefficients equal to the index minus one; Nonperiodicity of the telescoping example; Rationality forces some fixed shift to fail eventual smallness.

Further finite identities. Finite prime sums in the second normalisation; Adding one term to a dyadic partial sum; Adding one term to a difference partial sum; Consecutive primes are in increasing order; The natural-number gap agrees with subtraction in the rationals; Finite sums of weighted prime gaps; Finite sums for the telescoping coefficients; A nonnegative version of the telescoping coefficients; Agreement of the nonnegative and integer formulas.

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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