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

PDFOpen the PDF, 163 pages TeXLaTeX source Source on GitHub Problem page

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

Rational comparison sequences

Rational series can agree with the totient series on a fixed prefix, on parity at every index, or on specified linear data. The following constructions keep these requirements separate. Each gives a rational comparison satisfying the data named in its statement, and thus identifies what those data alone leave undetermined. The properties in different rows below belong to different constructions; they must not be combined into one comparison theorem.

Construction	Data kept while the value is rational
Prescribed prefix (Theorem 2.4)	Any fixed totient prefix and the bound $0 \leq c (n) \leq n$ .
Parity (Proposition 1.9)	Parity at every index, aperiodicity and the stated size bound.
Value $5 / 4$ (Section 12.6)	Every odd totient, changes of at most $2$ at even indices, and unbounded carry rank.
Signed interpolation (Section 10.10)	A prefix, all odd totients, and the specified moment and analytic data; not the full radial germ or all arithmetic identities.

Finite data and rational continuations

A rational sequence with any prescribed totient prefix

Limits of specific reductions and estimates

Consequences of the comparisons

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, 163 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:2be13b340f9dafc4. 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.