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

Multiples, diagonals, and LCM grids

Completeness allows the irrationality question to be expressed using multiples of a period, diagonal tails or an LCM grid. The implications below prove these equivalences. Each formulation permits a different estimate to be attempted, but none supplies the witnesses it requires.

Multiples of a period.

\forall h_{0} > 0, \forall N_{0}, \exists m > 0, \exists N \geq N_{0}, \exists L, C (m \cdot h_{0}, N, L) ⟹ S \notin Q .

Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_multiple_certificate_supply

The preceding residue condition implies this one by taking $m = 1$ . Conversely, a hypothetical period $h_{0}$ also makes every positive multiple $m h_{0}$ a period of the tail fractional parts. A certificate for one such multiple contradicts that consequence. The freedom to choose $m$ changes the allowed witnesses, while the quantified assertion remains equivalent to irrationality.

The diagonal.

S \notin Q ⟺ \forall t_{0}, \exists t \geq t_{0}, \exists L, C (H_{t}, H_{t}, L) .

Erdos249257.TotientTailPeriodKiller.irrational_totient_series_iff_lcm_diagonal_certificate_supply

The two parameters in a hypothetical rational tail period are absorbed by one LCM index. For $t \geq max (h_{0}, N_{0})$ , both $h_{0} ∣ H_{t}$ and $H_{t} \geq t \geq N_{0}$ hold. A diagonal certificate then contradicts periodicity through the multiple-period argument; the precise intermediate implication is proved in Lean (Erdos249257.TotientTailPeriodKiller.irrational_totient_series_of_lcm_certificate_supply). Conversely, irrationality and pointwise completeness give a diagonal certificate already at $t = t_{0}$ .

The finite assertion $P (t) :\equiv \exists L C (H_{t}, H_{t}, L)$ is verified for every $t \leq 82$ (ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82). The historical list contained 28 explicit values through $t = 64$ , with depths $[6, 5, 7, 7, 9, 14, 15, 14, 21, 22, 23, 26, \dots]$ for $t \in {1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, \dots}$ (Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_all_imported; endpoint Erdos249257.TotientTailPeriodKiller.certifiedKill_diagonal_t64). These are finite computations. The current contiguous range is still finite and does not establish witnesses at arbitrarily large indices; no certificate at $t = 83$ is claimed here.

The LCM grid.

\forall t_{0}, \exists t \geq t_{0}, \exists q, m, L, 0 < q \land C (m \cdot H_{t}, q \cdot H_{t}, L) ⟹ S \notin Q .

Erdos249257.irrational_totient_series_of_lcm_cone_window_kill_supply The diagonal choice is $q = m = 1$ . Allowing other positive grid points therefore enlarges the choice of finite witnesses. To prove sufficiency, suppose rationality supplies a period $h$ and pre-period $N_{0}$ . For $t \geq max (h, N_{0})$ , all positive multiples of $H_{t}$ are past the pre-period and their differences are multiples of $h$ . Consequently their tails have the same fractional part: $R_{q H_{t} + m H_{t}} - R_{q H_{t}} \in Z$ for every $q > 0$ and $m \geq 0$ (Erdos249257.rational_totient_series_forces_lcm_cone_flatness). A certificate for any such difference is a contradiction.

The finite-grid condition of Theorem 6.56 offers a second test. At $H = H_{t}$ , choose a finite nonempty $Q \subseteq N_{> 0}$ with $B_{q} = q H_{t} + L + 2 < 2^{L}$ for every $q$ . It requires

\forall q_{i} \in Q \exists q_{j} \in Q, B_{q_{j}} < (A_{q_{i}} - A_{q_{j}}) mod 2^{L}, A_{q} = \sum_{j = 1}^{L} φ (q H_{t} + j) 2^{L - j} .

For every threshold $t_{0}$ , such data at some $t \geq t_{0}$ would imply irrationality. This quantified condition, including the nonemptiness and size assumptions, remains unproved (Erdos249257.irrational_totient_series_of_lcm_cone_nonflat_supply). The modulus bound is one-sided at each point; it does not imply a factor-of-two reduction in the number of binary digits.

Equivalent nonintegrality statements

Via the completeness iff above, the diagonal and cone forms restate, exactly, with no reference to $C$ at all:

\forall t_{0}, \exists t \geq t_{0}, R_{2 H_{t}} - R_{H_{t}} \notin Z ⟺ diagonal certificate condition ⟹ S \notin Q,

Erdos249257.irrational_totient_series_of_lcm_diagonal_nonintegrality_supply (cone analogue at Erdos249257.irrational_totient_series_of_lcm_cone_nonintegrality_supply)

Thus one can ask directly whether $R_{2 H_{t}} - R_{H_{t}} \notin Z$ for arbitrarily large $t$ . Finite instances are known in this record; the universally quantified assertion is not proved here.

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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