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

PDFOpen the PDF, 163 pages TeXLaTeX source Source on GitHub Problem page

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.

Start here, by the editorial route

In this paper

Diagonal certificates through scale 82

Each row is a finite instance of Definition 1.2, with a link to the computation producing its witness. The aggregate result covers every integer scale $t \leq 82$ . It contains no assertion at $t = 83$ or at arbitrarily large scales.

Definition of the finite residue test

For $h, N, L : N$ , recall the truncated difference $D (h, N, L) := \sum_{j < L} (φ (N + h + 1 + j) - φ (N + 1 + j)) \cdot 2^{L - 1 - j} \in Z$ (windowDiscrepancy), and

C (h, N, L) :\equiv (N + h + L + 2) < D (h, N, L) mod 2^{L} < 2^{L} - (N + h + L + 2),

(certifiedKill; a decidable condition). By completeness (exists_certifiedKill_iff_tail_diff_notMem_int),

(\exists L, C (h, N, L)) ⟺ R_{N + h} - R_{N} \notin Z

for every $h, N$ . Requiring such certificates beyond every basepoint threshold for each fixed positive $h$ is precisely $Sep$ and is equivalent to $S \notin Q$ . The table records finite instances, indexed either by $(h, N, L)$ or by $t$ ; it does not establish that quantified condition.

Computation	Exact statement of what was checked	Lean source
Fixed-window example, depth 16	$C (h, 12, 16)$ holds for every $h \in {1, \dots, 8}$ (by direct evaluation, $256$ evaluations with totient arguments below $37$ ); consequently $S \neq r$ for every $r \in Q$ whose reduced denominator divides $2^{12} (2^{h} - 1)$ for some $1 \leq h \leq 8$ .	`certifiedKill_all_small` `totient_series_ne_rat_of_den_dvd`
Fixed-window example, depth 9, wider net	$C (h, 14, 9)$ holds for every $h \in {1, \dots, 16}$ : $S \neq r$ for every $r \in Q$ whose reduced denominator divides $2^{14} (2^{h} - 1)$ for some $1 \leq h \leq 16$ .	`totient_series_ne_rat_of_den_dvd_pow_two_mul_mersenne_upto_sixteen`
Diagonal certificates, 12 base scales	$C (H_{t}, H_{t}, L_{t})$ , $H_{t} := lcm (1, \dots, t)$ , holds (by explicit evaluation of the totient values, via factored prime-power blocks with Lucas-primality certificates) for $t \in {1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17}$ with certificate depths $L_{t} \in {6, 5, 7, 7, 9, 14, 15, 14, 21, 22, 23, 26}$ respectively (in the same order).	`certifiedKill_diagonal_all_imported` `diagonalPincerCertificateScales` `diagonalPincerKillDepth`
Diagonal certificates, further scales through $t = 64$	The same predicate $C (H_{t}, H_{t}, L_{t})$ is additionally checked, one Lean file per scale, at $t \in {19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64}$ (16 further explicit values). Together with the base 12 scales above this is 28 explicit values of $t$ in total, the complete list being ${1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64}$ . This does not establish $C (H_{t}, H_{t}, \cdot)$ for infinitely many $t$ ; these 28 examples are a historical strict subset of the contiguous band in the next row.	One Lean file per scale; endpoint `certifiedKill_diagonal_t64`
Contiguous lcm-diagonal band through $t \leq 82$	For every natural $t \leq 82$ there exists a depth $L$ with $C (H_{t}, H_{t}, L)$ . This closes every scale in the finite interval, including plateau transfers, with no holes. The next lcm jump is at the prime $83$ ; no certificate at $t = 83$ is claimed, and any such certificate must have depth at least $125$ .	`ErdosProblems.Skip.LadderT67.exists_diagonalKill_le_82` `ErdosProblems.Skip.LadderT67.t83_depth_floor`
Farey denominator bound	Every $q \in N$ with satisfies the window- $(N, K) = (1, 240)$ gap certificate; $q = Q_{0} + 1$ is the exact first failure.	`gap_check_window_1_240_le_79639646646701375323355774875831053`
Coprimality-probability / Möbius-square Farey bound	Every $d \in N$ with (exactly $⌊ Q_{0} / 2 ⌋$ ) is excluded as a denominator of $Pr (gcd (X, Y) = 1)$ and, separately, of $T = \sum_{d} μ (d) / (2^{d} - 1)^{2}$ .	`tsum_visible_coprime_pairs_ne_int_div_of_den_le_39819823323350687661677887437915526` `tsum_moebius_div_two_pow_sub_one_sq_ne_int_div_of_den_le_39819823323350687661677887437915526`

Remark 5.20 (Scope of the table). No row above is cofinal in its indexing parameter ( $h$ , $t$ , or the denominator bound), and no row is claimed to extrapolate. The historical diagonal depths are irregular and nonmonotone ( $6, 5, 7, 7, 9, 14, 15, 14, 21, 22, 23, 26, \dots$ ); no closed-form growth rate for $L_{t}$ is proved or conjectured in the record. This table records the historical 28-scale list through $t = 64$ ; the later aggregate theorem for every $t \leq 82$ is stated above and is not itemised row by row here. Neither finite record supplies the quantified condition of Definition 1.2.

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.

Prefer the manuscript?

PDFOpen the PDF, 163 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:2be13b340f9dafc4. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.