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.

Start here, by the editorial route

In this paper

The series

We begin with the value and the prefix-tail decomposition used by all the subsequent tests.

Definition 5.1 (The Erdős–249 constant). Let $φ$ denote Euler’s totient function. Define

S := \sum_{n \geq 1} \frac{φ (n)}{2^{n}} = \sum_{n : N} \frac{φ (n)}{2^{n}},

the two forms coinciding because $φ (0) = 0$ ; the second, $N$ -indexed form is the one carried in the Lean source. The series converges absolutely since $φ (n) \leq n$ . The irrationality of this value is the question under consideration.

Remark 5.2 (Status). No proof or disproof of $S \notin Q$ exists anywhere in the record, formal or informal. What exists is: one large unconditional finite denominator exclusion (§5.3), several exact reformulations of $S$ that relocate the same open question onto different coordinates without touching its truth value (§5.4–§5.6), and a certificate apparatus whose cofinal quantified condition is the precise open target (Definition 1.2; only its finite, checked instances are catalogued here, §5.7).

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.

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