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

The finite denominator bound

An interval containing $S$ excludes rational numbers whose denominators are too small to occur inside it. The Farey argument below makes this precise using a finite totient prefix and a bound on its tail. It is independent of the period and residue criteria.

Lemma 5.4 (Farey gap, fully general). For integers $a, b, c, d, r, s$ with $b > 0$ , $d > 0$ , $b c - a d = 1$ (i.e. $a / b$ and $c / d$ are unimodular Farey neighbours), and $a s < r b$ , $r d < c s$ (i.e. $r / s$ lies strictly between them): $b + d \leq s$ .

Indeed, the two strict inequalities make $c s - d r$ and $b r - a s$ positive integers, and

s = b (c s - d r) + d (b r - a s) \geq b + d,

using $b c - a d = 1$ . This also proves $s > 0$ , which the formulation did not need to assume separately. The lemma is the classical Stern–Brocot denominator bound. The mediant $(a + c) / (b + d)$ lies between the two neighbours and is reduced: any common divisor of $a + c$ and $b + d$ divides $b (a + c) - a (b + d) = b c - a d = 1$ . It attains denominator $b + d$ , so the general interval bound is sharp. A particular tail interval still requires its own endpoint and first-failure checks.

Proposition 5.5 (The Farey certificate at $K = 240$ ). For the window $(N, K) = (1, 240)$ , put

V = (\sum_{r = 1}^{240} φ (1 + r) 2^{240 - r}) mod 2^{240} .

For every $q : N$ with

0 < q \leq Q_{0} := 79 639 646 646 701 375 323 355 774 875 831 053 (\approx 7.96 \times 10^{34}),

(q \cdot V) mod 2^{240} + 243 q < 2^{240} .

This bound is sharp: $q = Q_{0} + 1 = 79 639 646 646 701 375 323 355 774 875 831 054$ is the exact first failing denominator, obtained as the denominator of the mediant of two explicit unimodular Farey neighbours.

Here “failing” means that this particular enclosure can no longer exclude the denominator. It does not mean that the mediant equals $S$ , or that another argument could not exclude that denominator too. Sharpness concerns the finite interval test, not a smallest possible denominator for the unknown value.

Theorem 5.6 (Denominator exclusion). For every $p \in Q$ whose reduced denominator is at most $Q_{0}$ ,

S \neq p .

Equivalently: if $S$ is rational, its reduced denominator exceeds $Q_{0} \approx 7.96 \times 10^{34}$ .

Remark 5.7 (What this does and does not say). Theorem 5.6 is a complete, unconditional finite fact: no rational of denominator at most $Q_{0}$ equals $S$ . A single such bound does not exclude arbitrarily large denominators. Proving unbounded exclusions by the same mechanism would, however, establish irrationality. The question this leaves is whether the first failing denominator $(b + d) (K)$ at window $K$ is unbounded as $K \to \infty$ . An affirmative answer would prove irrationality by Proposition 6.75, independently of the certificate condition discussed after Proposition 5.3. The question is open; it is neither claimed nor proved anywhere in the record.

Remark 5.8 (Earlier finite denominator bounds). The recorded Farey computations gave bounds $4838$ , $2^{22}$ , approximately $2.49 \times 10^{17}$ at $K = 120$ , and $Q_{0}$ at $K = 240$ . Each applies the finite mechanism of Lemma 5.4 and Proposition 5.5. These values do not prove that the bounds are unbounded as $K$ increases.

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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