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 of a period

We allow a positive multiple of each prescribed period. Telescoping proves sufficiency, and completeness of the residue test proves necessity, giving another condition equivalent to irrationality. The universally quantified condition remains unproved. For the finite examples, we specify which excluded denominators already occur in the diagonal computations.

Theorem 6.159 (Concatenation and specified period multiples). For nonnegative integers $a, b, N$ , define the integer block sum

Q_{a, N} = \sum_{j = 1}^{a} φ (N + j) 2^{a - j} .

Splitting a block after its first $a$ terms gives

Q_{a + b, N} = 2^{b} Q_{a, N} + Q_{b, N + a} .

In particular, $Q_{2 h, N} = 2^{h} Q_{h, N} + Q_{h, N + h}$ . The corresponding cyclotomic identity is $Φ_{4} (2^{h}) = 2^{2 h} + 1 = Φ_{2} (2^{2 h})$ : the order-four factor at height $h$ is the order-two factor at height $2 h$ .

The order-three factor is different: $Φ_{3} (2^{h}) = 2^{2 h} + 2^{h} + 1$ comes from tripling the period. It is not generally part of the doubling sequence. For example, $Φ_{3} (2) = 7$ divides none of $2^{2^{j}} - 1$ , because the order of $2$ modulo $7$ is $3$ , which does not divide $2^{j}$ . Thus the block identities relate specified period multiples; they do not put all the order-two, order-three and order-four factors into one doubling chain.

Observation 6.160 (An equivalent quantified condition). Both implications of this equivalence are proved in Lean (ErdosProblems.Erdos249.PeriodMultipleEscape.periodMultipleKillSupply_iff_irrational); the quantified condition itself is not established. The aggregate diagonal table gives certificates at every $H_{t}$ for $t \leq 82$ (ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82); this equivalence contributes no new information about $t = 83$ or any condition at arbitrarily large scales.

Observation 6.161 (Eight finite period certificates). The cited declarations give certificates at basepoint $N = 300$ for

h \in {67, 81, 97, 101, 121, 125, 127, 128} .

Each excludes every reduced fraction $a / b$ , with $b > 0$ and $b ∣ 2^{300} (2^{h} - 1)$ . These exclusions overlap with those already obtained from the diagonal. In particular, $67$ and $81$ divide $H_{82}$ , and $H_{82} > 300$ , so

2^{300} (2^{h} - 1) ∣ 2^{H_{82}} (2^{H_{82}} - 1) (h = 67, 81) .

Their entire denominator classes are already covered by Proposition 6.104. The eight cases therefore cannot all be described as new denominator classes relative to that proposition. The exact comparison of the odd factors is

gcd (2^{h} - 1, 2^{H_{82}} - 1) = 2^{gcd (h, H_{82})} - 1,

by the Euclidean algorithm on the exponents. None of the remaining six periods divides $H_{82}$ , so its full odd factor is not covered by that diagonal denominator. This compares with the $t = 82$ certificate, not with every other exclusion in the record; the case $h = 127$ includes the Mersenne prime $2^{127} - 1$ .

The case $h = 67$ also succeeds at depth $L = 67$ . It illustrates the restriction $L = h$ in the full-depth condition, while retaining a small basepoint. That universally quantified condition is equivalent to irrationality; one finite instance is not. ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300 ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_127_300 ErdosProblems.Erdos249.PeriodMultipleEscape.certifiedKill_67_300_fullDepth ErdosProblems.Erdos249.PeriodMultipleEscape.totient_series_ne_rat_of_den_dvd_127_300 These are eight finite exclusions; the condition at arbitrarily large scales remains 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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