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

A rational series with matching totient parity

Matching totient parity and the bound $c (n) \leq n$ does not determine the rationality of a binary sum. The example below is bounded and aperiodic, yet its sum is rational. Its paired changes can be placed beyond any threshold with any prescribed minimum separation.

Construction. Start with the sequence $0, 1, 1, 2, 4, 4, 4, \dots$ . At each position $m = 2^{k + 3}$ , $k \geq 0$ , replace the pair $(4, 4)$ at $m, m + 1$ by $(6, 0)$ . Equivalently,

c (n) = {\begin{cases} 0, & n = 0, \\ 1, & n = 1, 2, \\ 2, & n = 3, \\ 6, & n = 2^{k + 3} for some k \geq 0, \\ 0, & n = 2^{k + 3} + 1 for some k \geq 0, \\ 4, & otherwise . \end{cases}

Each replacement leaves the binary sum unchanged because $2 \cdot 2^{- m} - 4 \cdot 2^{- (m + 1)} = 0$ . The increasingly large gaps between replacements prevent eventual periodicity. The strongest recorded result also quantifies their separation:

Proposition 10.1 (A rational series preserving totient parity and the stated separation properties). There exists $c : N \to N$ such that: $c (n) \leq 6$ for all $n$ ; $c (n) \leq n$ ; $c (n) \equiv φ (n) (\mod 2)$ for every $n$ ; for every $N, G, K$ there are $K$ pairs of coefficients $(6, 0)$ starting beyond $N$ , with starting positions separated by more than $G$ ; and $\sum_{n} c (n) / 2^{n} = 3 / 2$ .

Scope. The displayed bounds, parity agreement and aperiodicity, including the stated number and separation of the edits, are jointly compatible with a rational binary sum. They therefore do not imply irrationality. The sequence $c$ differs from $φ$ : the example preserves its parity, not all its arithmetic properties. It excludes an argument using only the listed assumptions, not every strengthening of an aperiodicity condition or every use of totient coefficients.

Consequence. The paired edits have zero total contribution, since $2 \cdot 2^{- m} - 4 \cdot 2^{- (m + 1)} = 0$ . Applied at the specified sparse positions of this particular base sequence, they preserve the rational sum and the required bounds while preventing eventual periodicity. This does not construct a rational series for every prescribed parity pattern, and it is not the finite-prefix construction of Theorem 2.4. A proposed criterion using only the assumptions of this example must fail; a criterion using further arithmetic information is not addressed.

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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