Running least common multiples: distinct values and multiplicities

Will Cook

Long paper

Running least common multiples: distinct values and multiplicities

Erdős #269 By Will Cook July 2026; revised 5 October 2026 Authorship, AI use and citation

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Summary. For the prime set {2,3,5}, a five-map argument proves irrationality of the sum over distinct running LCMs. Two-prime sums reduce to a transcendental Hecke–Mahler value. For the repeated three-prime sum, an equivalent residue criterion leaves an arithmetic condition open. Complete ordinary arguments for general distinct-running-LCM sums and prime-power subseries are retained as remarks awaiting formalisation.

This paper covers distinct-height irrationality, all-rank kernel structure and the repeated-sum tail criteria, preserving the multiplicity distinction and the remaining open residue-window obligation.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open.

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In this paper

A decoding criterion for every finite set of primes

For other prime sets, a useful tail interval may depend on the recent occurrence of each prime. Put $Q = P ∖ {p}$ and choose the integer $g_{q}$ by $p^{g_{q}} < q < p^{g_{q} + 1}$ . Successive blocks containing a $q$ -power have indices $⌊ f \log_{p} q ⌋$ and $⌊ (f + 1) \log_{p} q ⌋$ , by Remark 5.6(i). Their difference is $g_{q}$ or $g_{q} + 1$ . The gaps give a finite memory of the last occurrences. We use it to enclose tails of every allowed path, a class containing all sufficiently late arithmetic paths but possibly others. Separation on this larger class suffices for recovery; overlap need not give equal arithmetic tails. The memory sharpens the enclosures, but the decoder must ultimately work without being told the state.

These constraints define a finite automaton. Its states are the tuples $s = (s_{q})_{q \in Q}$ with $1 \leq s_{q} \leq g_{q} + 1$ ; the entry $s_{q}$ counts the blocks since the last block containing a power of $q$ . A transition from $s$ is a pair $(τ, s^{'})$ in which $τ$ is an ordering of a set $T \subseteq Q$ with

{q : s_{q} = g_{q} + 1} \subseteq T \subseteq {q : s_{q} \geq g_{q}},

and $s_{q}^{'} = 1$ for $q \in T$ , $s_{q}^{'} = s_{q} + 1$ for $q \notin T$ . A path from $s$ is a sequence of transitions $(u_{1}, s_{1}), (u_{2}, s_{2}), \dots$ with $(u_{1}, s_{1})$ a transition from $s$ and each $(u_{i + 1}, s_{i + 1})$ a transition from $s_{i}$ . The value of an infinite path is the limit of $G_{u_{1}} \circ \dots \circ G_{u_{n}} (y)$ as $n \to \infty$ ; by Remark 5.6(ii) these numbers lie, for $y \in [0, 1 / (p - 1)]$ , in nested intervals of length at most $p^{- n} / (p - 1)$ , so the limit exists and does not depend on $y$ .

Every prime $q \in Q$ lies in block $g_{q}$ . For $a > max_{q} g_{q}$ let $s_{a}$ record, for each $q$ , the number $a - ℓ$ , where $ℓ < a$ is the index of the last block before $a$ containing a power of $q$ . The next such block has index $ℓ + g_{q}$ or $ℓ + g_{q} + 1$ , and it is at least $a$ . Hence $s_{a}$ is a state, block $a$ contains a power of $q$ when $(s_{a})_{q} = g_{q} + 1$ and does not when $(s_{a})_{q} < g_{q}$ , and $(τ_{a}, s_{a + 1})$ is a transition from $s_{a}$ . Since $Y_{a} = G_{τ_{a}} \circ \dots \circ G_{τ_{a + n - 1}} (Y_{a + n})$ with $Y_{a + n} \in [0, 1 / (p - 1)]$ , the tail $Y_{a}$ is the value of the actual path $(τ_{a}, s_{a + 1}), (τ_{a + 1}, s_{a + 2}), \dots$ from $s_{a}$ .

Remark 5.7 (a decoding criterion). Let $P$ be a finite set of at least two primes, and for every state $s$ let $J (s)$ be a closed interval containing the values of all infinite paths from $s$ . For $k \geq 1$ and each type $τ$ , let $U_{τ}$ be the union of the intervals $G_{u_{1}} \circ \dots \circ G_{u_{k}} (J (s_{k}))$ over all states $s$ and all paths $(u_{1}, s_{1}), \dots, (u_{k}, s_{k})$ from $s$ with $u_{1} = τ$ . If the sets $U_{τ}$ are pairwise disjoint, then $D_{P}$ is irrational.

The union defining $U_{τ}$ runs over all starting states. Thus membership can identify the first block from $Y_{a}$ alone; separating types only within each known state would not give this conclusion.

Proof. Let $a > max_{q} g_{q}$ . The first $k$ transitions of the actual path from $s_{a}$ form a path of length $k$ with first type $τ_{a}$ , and $Y_{a} = G_{τ_{a}} \circ \dots \circ G_{τ_{a + k - 1}} (Y_{a + k})$ , where $Y_{a + k} \in J (s_{a + k})$ because it is the value of an infinite path from $s_{a + k}$ . Thus $Y_{a} \in U_{τ_{a}}$ , and by disjointness $τ_{a}$ is the only type with this property. So $Y_{a}$ determines $τ_{a}$ , then $Y_{a + 1} = b_{τ_{a}} Y_{a} - μ_{τ_{a}}$ , and by induction the word $τ_{a} τ_{a + 1} \dots$ .

Suppose $D_{P} = N / K$ . By Remark 5.6(ii) and (iii), the numbers $K Y_{a}$ are integers in $[0, K / (p - 1)]$ , so $Y_{a} = Y_{a^{'}}$ for some $max_{q} g_{q} < a < a^{'}$ , and the word $(τ_{a})$ is eventually periodic. Fix $q \in Q$ and put $β = \log p / \log q$ , which lies in $(0, 1)$ and is irrational because no power of $p$ equals a power of $q$ . By Remark 5.6(i), block $a$ contains $⌊ (a + 1) β ⌋ - ⌊ a β ⌋$ powers of $q$ , a number recorded by $τ_{a}$ . As in the proof of Theorem 1.1, eventual periodicity with period $ℓ$ forces $ℓ β \in Z$ , a contradiction. ◻

Intervals $J (s)$ with the required property are easy to produce. The constant interval $[0, 1 / (p - 1)]$ qualifies by Remark 5.6(ii). If a family $J$ qualifies, so does any family $J^{'}$ with

J^{'} (s) \supseteq G_{τ} (J (s^{'})) for every transition (τ, s^{'}) from s,

because an infinite path from $s$ is a transition $(τ, s^{'})$ followed by an infinite path from $s^{'}$ . Starting from the constant family, take the interval hull of the images and round its endpoints outwards. Every round preserves containment, though a hull may fill genuine gaps. Keep the separate image intervals when forming the final unions $U_{τ}$ . The smallest enclosing interval at state $s$ is $J (s) = [L (s), U (s)]$ , where $L (s)$ and $U (s)$ are the least and greatest infinite-path values. They satisfy $L (s) = min G_{τ} (L (s^{'}))$ and $U (s) = max G_{τ} (U (s^{'}))$ over the transitions from $s$ , and these equations have unique solutions because each $G_{τ}$ contracts by at least the factor $1 / p$ .

For ${2, 3, 5}$ the constant interval $I = [3 / 10, 13 / 15]$ has the required property: every path takes a transition with $5 \in T$ within any three steps, which is the only fact the bounds used. With $k = 1$ the criterion is then the proof given above.

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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