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

Residues modulo divisors of Mersenne numbers

Fix a hypothetical denominator $2^{c} v$ , with $c \geq 0$ and odd $v \geq 1$ . For a positive multiple $H$ of $φ (v)$ , Euler’s theorem makes

M = \frac{2^{H} - 1}{v}, B_{H, c} = \sum_{j = 0}^{H - 1} φ (c + 1 + j) 2^{H - 1 - j}

an integer modulus and an integer prefix, respectively. The test below asks whether the residue of $- B_{H, c}$ stays away from the endpoints by more than the tail error.

Equivalent formulation A.2 (A residue test modulo a divisor of $2^{H} - 1$ ). The series $S$ is irrational if and only if, for every $c \geq 0$ and odd $v \geq 1$ , some positive multiple $H$ of $φ (v)$ satisfies

\begin{matrix} (8) & c + H + 1 < (- B_{H, c}) mod M < M - (c + H + 1) . \end{matrix}

Proof. The identity

B_{H, c} = 2^{H} R_{c} - R_{c + H} = M (v R_{c}) - Δ_{H} (c)

is exact. For $H \geq 1$ , $| Δ_{H} (c) | < c + H + 1$ . Indeed $R_{N} \leq N + 1$ for $N \geq 1$ and $0 < R_{0} \leq 3 / 2$ ; the strict inequalities follow by subtracting a positive tail. If $S = a / (2^{c} v)$ , then $v R_{c} \in Z$ and $Δ_{H} (c) \in Z$ . The residue of $- B_{H, c}$ is therefore within distance less than $c + H + 1$ of $0 (\mod M)$ , so (8) fails.

If $S$ is irrational, fix $c, v$ and put $δ = dist (v R_{c}, Z) > 0$ . As $H$ grows through positive multiples of $φ (v)$ , choose it large enough that $| Δ_{H} (c) | / M < δ / 3$ and $(c + H + 1) / M < δ / 3$ . Since $- B_{H, c} / M = - v R_{c} + Δ_{H} (c) / M$ , its distance from $Z$ is then greater than $2 δ / 3$ , while the forbidden endpoint width is below $δ / 3$ . This proves (8) for all sufficiently large allowed $H$ . ◻

For fixed $c, v$ , one successful test excludes every value $S = a / (2^{c} v)$ , regardless of the numerator $a$ . Conversely, irrationality gives success for every sufficiently large allowed $H$ for each fixed $c, v$ . This converse does not supply successful tests for every $c, v$ without the irrationality assumption.

A formal declaration for the equivalence is available here, with the modulus written as $v M = 2^{H} - 1$ . The finite residues also satisfy

ρ_{H, N + 1, M} = (2 ρ_{H, N, M} - φ (N + H + 1) + φ (N + 1)) mod M, ρ_{H, N, M} = (- B_{H, N}) mod M,

which is an exact recurrence for searches directed at (8).

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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