The Binary Totient Series

Will Cook

Long paper

The Binary Totient Series

Erdős #249 By Will Cook July 2026; revised 1 October 2026 Authorship, AI use and citation

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Summary. The dyadic totient sections through level $e \geq 1$ have an explicit rational basis of size $2^{e} + 1$ . Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ $φ (n) 2^{- n}$ .

This paper covers a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

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In this paper

Prime divisors of Mersenne factors and finite exclusions

For $X - 2$ , multiplicative orders supply infinitely many prime divisors among the Mersenne factors. They therefore satisfy the abstract prime-support hypothesis below. This does not imply the required tail-residue inequality. The accompanying finite exclusions concern only their stated periods and basepoints.

Theorem 6.151 (Unbounded prime support in Mersenne factors). Every prime divisor $p$ of $2^{q} - 1$ , for prime $q$ , satisfies $q ∣ p - 1$ (the order of $2$ mod $p$ is exactly $q$ , by Fermat/Lagrange in $(Z / p)^{\times}$ ); consequently the prime divisors appearing in the layers ${2^{n} - 1}$ are unbounded, unconditionally, with no cyclotomic resultant hypothesis left open.

Theorem 6.152 (A prime satisfying the stated cyclotomic conditions). For every period $h > 0$ and threshold $N_{0}$ , there exist a prime $q$ and a prime factor $p$ of $| Φ_{h q} (2) |$ (the binary cyclotomic layer) with $p$ coprime to $h q$ , $h q ∣ p - 1$ , and $p - 1 \geq N_{0}$ . The characteristic-prime exceptional case in the cyclotomic order decomposition is eliminated directly, by choosing $q > 2^{h}$ (rules out $p = q$ ) and $q > h$ (rules out $p ∣ h$ ).

Observation 6.153. The first theorem uses the multiplicative order of $2$ modulo a prime. For the second, choose a prime $q > max (h, N_{0}, 2^{h})$ and then a prime divisor of $| Φ_{h q} (2) | > 1$ . The cyclotomic order formula, together with the stated inequalities on $q$ , excludes prime divisors of $h q$ and gives ${ord}_{p} (2) = h q$ . Existence of arbitrarily large primes suffices here; no prime-in-progression theorem is needed for this construction.

These results supply prime divisors with the required orders. They do not supply the tail-residue inequality at the resulting periods. The all-prime and eventual forms of the order hypothesis must also be distinguished, as explained in §6.6.12.

Observation 6.154 (Finite exclusions at period 30). The Lean kernel decides two independent certificates at the composite period $h = 30$ , at the basepoints $N = 300$ and $N = 330$ . The certificate at $N = 300$ gives the denominator exclusion $S \neq r$ for every $r \in Q$ whose reduced denominator divides $2^{300} (2^{30} - 1)$ . ErdosProblems.Erdos249.CyclotomicAnchoredKill.certifiedKill_cyclotomic_30_331_natural ErdosProblems.Erdos249.CyclotomicAnchoredKill.totient_series_ne_rat_of_den_dvd_30_300 This is a finite exclusion; the corresponding condition at arbitrarily large scales remains unproved.

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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