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 single-parameter diagonal condition

Taking the period to be a least common multiple reduces the certificate condition to a single scale $t$ . The witness must still occur beyond every threshold in $t$ . We prove the reduction and retain that arithmetic requirement explicitly.

Observation 6.148 (The diagonal implication). The irrationality of $S$ follows if, for every integer $t_{0} \geq 0$ , there are $t \geq t_{0}$ and $L \geq 0$ such that $C (H_{t}, H_{t}, L)$ , where $H_{t} = lcm (1, \dots, t)$ and $H_{0} = 1$ . This implication is proved in Lean (Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_lcm_diagonal_certificate_supply); its hypothesis is unproved. Theorem 6.49 also records the converse. The quantified condition is therefore equivalent to irrationality, not a weaker hypothesis.

Theorem 6.149 (Nondivisors in a short LCM window). Let $t \geq 1$ . If $1 \leq j < 2 t$ and $j ∤ H_{t}$ , then $j = p^{a} > t$ for a prime $p$ and an integer $a \geq 1$ . Every integer $1 \leq j \leq t$ divides $H_{t}$ . To see the first claim, some prime-power divisor $p^{a}$ of $j$ exceeds $t$ ; otherwise every prime-power divisor would divide $H_{t}$ . Since $j < 2 t < 2 p^{a}$ , its remaining cofactor is $1$ .

If $j ∣ H_{t}$ and every prime divisor of $j$ also divides $H_{t} / j$ , then, for every integer $q \geq 0$ ,

φ (q H_{t} + j) = φ (j) φ (q (H_{t} / j) + 1) .

Indeed, the second factor’s argument is coprime to $j$ , so totient multiplicativity applies. The hypothesis is essential to this factorisation: at $t = 2$ , $j = 2$ , $q = 1$ the left side is $φ (4) = 2$ but the displayed product would be $1$ .

Observation 6.150 (Finite diagonal examples). The cited file proves $C (H_{t}, H_{t}, L)$ at the following depths:

\begin{array}{crrrrrrrr} t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ H_{t} & 1 & 2 & 6 & 12 & 60 & 60 & 420 & 840 \\ L & 6 & 5 & 7 & 7 & 9 & 9 & 14 & 15 \end{array}

The largest totient argument is $2 H_{6} + 9 = 129$ for $t \leq 6$ , and $2 H_{8} + 15 = 1695$ for the full table. Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_upto_six Erdos249257.TotientTailPeriodKiller.certifiedKill_periodLcm_diagonal_at_eight This is a finite verification; certificates at arbitrarily large scales remain unproved.

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.

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