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

Equivalent quantified certificate conditions

Completeness identifies the base residue condition and the LCM-diagonal condition with irrationality. The following statements spell out the quantifiers and the implications between them. The existence of the required certificates beyond every threshold remains unproved.

Theorem 6.47 (The quantified certificate condition is equivalent to irrationality). $(\forall h : N, 0 < h \to \forall N_{0} : N, \exists N \geq N_{0}, \exists L, C (h, N, L)) \leftrightarrow S \notin Q$ . The quantified-condition side is exactly $C (h, N, L)$ quantified as $\forall h \geq 1 \forall N_{0} \geq 0 \exists N \geq N_{0} \exists L$ . The universally quantified assertion remains unproved. The equivalence identifies the finite witnesses that suffice; it does not establish their existence beyond every threshold.

Theorem 6.48 (It suffices to use multiples of a period). $(\forall h_{0} > 0, \forall N_{0}, \exists m > 0, \exists N \geq N_{0}, \exists L, C (m \cdot h_{0}, N, L)) \to S \notin Q$ . The shift may be any positive multiple of a prescribed period. This enlarges the choice of finite witness: Theorem 6.47 gives the condition with $m = 1$ . Conversely, rationality would make all such tail differences integral after a fixed starting index. The displayed implication therefore makes this quantified condition equivalent to irrationality; its truth remains unproved.

Theorem 6.49 (The diagonal condition is equivalent to irrationality). $(\forall t_{0} : N, \exists t \geq t_{0}, \exists L, C (H (t), H (t), L)) \leftrightarrow S \notin Q$ . For a fixed hypothetical rational value, sufficiently large $t$ makes both its dyadic denominator and its odd-part period admissible at $N = h = H (t)$ . Conversely, irrationality and pointwise completeness supply a witness for every prescribed $t$ . This is an equivalent condition with one scale parameter; the verified cases $t \leq 82$ do not establish it at arbitrarily large scales.

Lemma 6.50 (Nondivisors in a short LCM window). Let $t \geq 1$ and $1 \leq j < 2 t$ be integers. If $j ∤ H (t)$ , then $j = p^{a} > t$ for a prime $p$ and an integer $a \geq 1$ . Indeed, a prime-power divisor of $j$ must exceed $t$ , and $j < 2 t$ leaves no room for a cofactor larger than $1$ . This classifies the exceptional offsets in this short window; it does not bound their contribution to a weighted sum of totient differences.

Proposition 6.51 (Totient factorisation on an LCM progression). Let $t \geq 1$ , $j \geq 1$ and $q \geq 0$ be integers. Suppose $j ∣ H (t)$ and every prime divisor of $j$ also divides $H (t) / j$ . Then

φ (q H (t) + j) = φ (j) φ (q (H (t) / j) + 1) .

The two factors in $q H (t) + j = j (q (H (t) / j) + 1)$ are coprime: the second is $1$ modulo each prime dividing $j$ . Totient multiplicativity gives the identity. The extra prime-divisor hypothesis is not automatic from $j ∣ H (t)$ ; for example, it fails at $t = j = 2$ , where $φ (H (2) + 2) = 2$ but the proposed product would be $1$ .

Theorem 6.52 (Rationality forces tail integrality on an LCM grid). If $S \in Q$ , there is $t_{1} \in N$ such that for all $t \geq t_{1}$ , $q \geq 1$ and $m \geq 0$ ,

R_{(q + m) H (t)} - R_{q H (t)} \in Z .

Thus all tails at positive multiples of $H (t)$ have the same fractional part once $t \geq t_{1}$ . A certificate at one fixed scale does not refute rationality: the threshold $t_{1}$ depends on the hypothetical rational value. At a shift $m H (t) > 0$ and basepoint $q H (t)$ , such a certificate excludes rational values whose reduced denominator divides $2^{q H (t)} (2^{m H (t)} - 1)$ . Certificates at arbitrarily large scales, as required in Theorem 6.53, exclude every hypothetical rational value.

Theorem 6.53 (A sufficient nonintegrality condition on the grid). Suppose that for every $t_{0} \in N$ there are integers $t \geq t_{0}$ , $q \geq 1$ , $m \geq 0$ and $L \geq 0$ such that

C (m H_{t}, q H_{t}, L) .

Then $S \notin Q$ . Rationality would make the corresponding tail difference integral at every sufficiently large scale, contradicting Proposition 6.18. A successful certificate necessarily has $m > 0$ , since the discrepancy vanishes at shift zero.

The diagonal choice is $q = m = 1$ ; consecutive grid points have $m = 1$ , and for $p \geq 2$ the pair $(H_{t}, p H_{t})$ has $(q, m) = (1, p - 1)$ . The diagonal equivalence in Theorem 6.49 also gives the converse from irrationality to the displayed quantified condition; that condition remains unproved.

Corollary 6.54 (Equivalent conditions stated without certificates). The series $S$ is irrational if and only if

R_{2 H_{t}} - R_{H_{t}} \notin Z for arbitrarily large integers t .

Equivalently, for arbitrarily large $t$ there are integers $q \geq 1$ and $m \geq 1$ with $R_{(q + m) H_{t}} - R_{q H_{t}} \notin Z$ . Apply the pointwise equivalence of Theorem 6.19 to Theorems 6.49 and 6.53, respectively. These are equivalent statements of the irrationality question; the required unbounded sets of scales are not established.

Proposition 6.55 (Soundness of a second-difference certificate). Let $h, N, L \in N$ . If

\begin{aligned} 2 (N + 2 h + L + 2) & < (D (h, N + h, L) - D (h, N, L)) mod 2^{L} \\ < 2^{L} - 2 (N + 2 h + L + 2), \end{aligned}

then $R_{N + 2 h} - 2 R_{N + h} + R_{N} \notin Z$ . The numerator is the second difference of the finite windows. Subtracting the two tail identities leaves an error of absolute value at most $2 (N + 2 h + L + 2)$ , which explains the radius in the hypothesis. At $(h, N) = (1, 8)$ , the first-difference test holds at depth $8$ , whereas the second-difference test fails at every depth $L \leq 8$ and holds at $L = 9$ . A historical probe over $t \leq 20$ reported that the first-difference test was at least as shallow in $30$ of $40$ sampled cases. This finite comparison is not a uniform ordering of the depths.

For $H, L \in N$ and a finite set $Q$ of multipliers, put

A_{q} = \sum_{j = 1}^{L} φ (q H + j) 2^{L - j}, B_{q} = q H + L + 2 (q \in Q) .

The finite-grid test compares these integer numerators modulo $2^{L}$ .

Theorem 6.56 (A finite-grid condition gives a nonintegral pair). Let $Q \subseteq N_{> 0}$ be finite and nonempty. Suppose that $B_{q} < 2^{L}$ for every $q \in Q$ and that

\forall q_{i} \in Q \exists q_{j} \in Q, B_{q_{j}} < (A_{q_{i}} - A_{q_{j}}) mod 2^{L} .

Then $R_{q_{j} H} - R_{q_{i} H} \notin Z$ for some $q_{i}, q_{j} \in Q$ .

Proof. The tail identity and its elementary bound give

2^{L} R_{q H} = A_{q} + R_{q H + L}, 0 \leq R_{q H + L} \leq B_{q} .

Suppose all pairwise tail differences are integers. Choose $q_{*} \in Q$ minimising $R_{q H + L}$ . For every $q \in Q$ , the difference $R_{q H + L} - R_{q_{*} H + L}$ is then an integer in $[0, B_{q}]$ , congruent to $A_{q_{*}} - A_{q}$ modulo $2^{L}$ . Since $B_{q} < 2^{L}$ , it is the least nonnegative residue. This contradicts the displayed condition at $q_{*}$ . ◻

For example, take $H = 840$ , $L = 12$ and $Q = {1, 2, 3, 4}$ . The following exact values verify one failed arc containment in each row:

$q_{i}$	$A_{q_{i}} mod 4096$	$B_{q_{i}}$	$q_{j}$	$(A_{q_{i}} - A_{q_{j}}) mod 4096$
1	2920	854	3	3484
2	2024	1694	1	3200
3	3532	2534	4	3832
4	3796	3374	1	876

The last column exceeds $B_{q_{j}}$ , not necessarily $B_{q_{i}}$ . On the circle $R / 4096 Z$ , every two arcs $[A_{q}, A_{q} + B_{q}]$ intersect, but the four arcs have no common point. Thus a test using all four points can exclude a common fractional part without any pair of these arcs being disjoint. The theorem does not require pairwise compatibility, or require $| Q | \geq 3$ .

The size hypothesis is $2^{L} > max_{q \in Q} B_{q}$ . In contrast, the symmetric test $C$ for a pair $q_{i} < q_{j}$ needs $2^{L} > 2 B_{q_{j}}$ before its two-sided interval can contain a residue. These are inequalities for the modulus $2^{L}$ . They do not say that the new test uses half as many binary digits, or give a uniform comparison of the first depths at which the tests succeed.

Theorem 6.57 (A sufficient quantified finite-grid condition). Suppose that for every $t_{0}$ there are $t \geq t_{0}$ , a depth $L$ , and a finite nonempty $Q \subseteq N_{> 0}$ such that, with $H = H_{t}$ , all the hypotheses of Theorem 6.56 hold. Then $S \notin Q$ . Indeed, rationality would make all tail differences on every sufficiently large LCM grid integral, whereas the finite-grid theorem supplies a nonintegral pair on such a grid. The example above verifies one grid; it does not establish the quantified hypothesis.

Proposition 6.58 (A finite carry test implies nonintegrality). For each integer $z$ with $| z | \leq N + h + 1$ , define

c_{z} (0) = z, c_{z} (i + 1) = 2 c_{z} (i) - (φ (N + h + i + 1) - φ (N + i + 1)) .

Suppose that for every such $z$ there is an $i \leq K$ with

| c_{z} (i) | \geq N + i + h + 2.

Then $R_{N + h} - R_{N} \notin Z$ . Indeed, an integral value of the tail difference would be one of the $2 (N + h + 1) + 1$ initial candidates. Its recurrence would satisfy $c_{z} (i) = R_{N + h + i} - R_{N + i}$ and hence $| c_{z} (i) | < N + i + h + 2$ at every step, contrary to the finite test. The excluded strip is open: equality at its boundary already suffices. This proposition proves soundness for a specified $h, N, K$ ; an irrationality argument still needs the quantified family of tests.

Theorem 6.59 (The block norm condition implies irrationality). Suppose that for every integer $h \geq 1$ and every threshold $X_{0} \in N$ there are $X, L \in N$ with

X \geq max (X_{0}, 1), 16 (2 X + h + L + 2) \leq 2^{L},

and

‖ \sum_{X \leq N < 2 X} E (h, N, L) ‖ \leq \frac{21}{25} X .

Then $S \notin Q$ . Indeed, the norm bound implies the real-part bound $Re \sum E (h, N, L) \leq (9 / 10) X$ . Under the displayed size condition, this latter bound already gives a finite residue certificate, by $\cos (π / 8) > 9 / 10$ and averaging. The argument uses this specific numerical gap; an unspecified saving below $X$ is not the stated hypothesis.

Theorem 6.60 (A four-tail residue criterion). For nonnegative integers $H, p, L$ , define the four-tail combination, its finite numerator and its error bound by

\begin{aligned} J (H, p) & = R_{2 p H} - R_{p H} - p R_{2 H} + p R_{H}, \\ W (H, p, L) & = D (p H, p H, L) - p D (H, H, L), \\ B (H, p, L) & = 3 p H + (p + 1) (L + 2) . \end{aligned}

Then

| 2^{L} J (H, p) - W (H, p, L) | \leq B (H, p, L) .

Consequently the finite condition

B (H, p, L) < W (H, p, L) mod 2^{L} < 2^{L} - B (H, p, L)

implies $J (H, p) \notin Z$ . If, for every $t_{0} \in N$ , there are $t \geq t_{0}$ , $p \geq 1$ and $L \geq 0$ satisfying this condition with $H = H (t)$ , then $S \notin Q$ . Rationality would instead make both diagonal differences in $J (H (t), p)$ integral for every sufficiently large $t$ .

The error bound groups the remainder as

(R_{2 p H + L} + p R_{H + L}) - (R_{p H + L} + p R_{2 H + L}) .

Each parenthesis is nonnegative and at most $B (H, p, L)$ , by $0 \leq R_{n} \leq n + 2$ . Their difference therefore has absolute value at most $B (H, p, L)$ . This proves the bound without asserting optimality for the actual totient tails. Primality of $p$ is not required by the criterion; a prime $p$ is useful when comparing consecutive LCM heights. The checked instance is $(H, p, L) = (12, 5, 15)$ , for which $W (H, p, L) mod 2^{L} = 18834$ and $B (H, p, L) = 282$ .

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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