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

Blocks and affine maps

We now extend the calculation of Section 2 to an arbitrary finite prime set.

To obtain a second description of the sum over distinct running LCMs, group its jumps into $(p^{a}, p^{a + 1}]$ , where $p$ is the least prime of $P$ , and define

P_{a} = H_{P} (p^{a}), Y_{a} = P_{a} \sum_{t_{k} > p^{a}} \frac{1}{H_{P} (t_{k})} .

Write $τ_{a} = (r_{1}, \dots, r_{v})$ for the labels of the jumps strictly inside block $a$ , in increasing order of their powers. This is the block’s type; write $\emptyset$ when there is no interior jump. The final jump at $p^{a + 1}$ always has label $p$ .

Remark 5.6 (blocks, tails and integrality). Let $P$ be a finite set of at least two primes with least element $p$ .

Block $a$ ends with the jump at $p^{a + 1}$ and contains at most one power of each prime $q \in P ∖ {p}$ , lying strictly inside it. The power $q^{f}$ with $f \geq 1$ lies in block $⌊ f \log_{p} q ⌋$ .
If block $a$ has type $τ = (r_{1}, \dots, r_{v})$ , then $Y_{a} = G_{τ} (Y_{a + 1})$ , where
$G_{τ} (y) = \frac{μ_{τ} + y}{b_{τ}}, b_{τ} = p r_{1} \dots r_{v}, μ_{τ} = 1 + b_{τ} \sum_{j = 1}^{v} \frac{1}{r_{1} \dots r_{j}} .$
Each $G_{τ}$ is increasing with slope $1 / b_{τ} \leq 1 / p$ and maps $[0, 1 / (p - 1)]$ into itself, and $0 < Y_{a} \leq 1 / (p - 1)$ .
If $D_{P} = N / K$ with integers $N$ and $K \geq 1$ , then $K Y_{a}$ is an integer for every $a \geq 0$ .

Proof. (i) The endpoint $p^{a + 1}$ is a jump. Consecutive powers of $q$ have ratio $q > p$ , so the block contains at most one of them, and no power of $q$ equals a power of $p$ . The power $q^{f}$ lies in $(p^{a}, p^{a + 1})$ exactly when $a < f \log_{p} q < a + 1$ .

(ii) The jumps of block $a$ are the interior powers, with running LCMs $P_{a} r_{1}$ , $P_{a} r_{1} r_{2}$ , …, $P_{a} r_{1} \dots r_{v}$ , followed by the jump at $p^{a + 1}$ , whose running LCM is $P_{a + 1} = b_{τ} P_{a}$ . Splitting off the block,

Y_{a} = \sum_{j = 1}^{v} \frac{1}{r_{1} \dots r_{j}} + \frac{1}{b_{τ}} + \frac{P_{a}}{P_{a + 1}} Y_{a + 1} = \frac{μ_{τ} + Y_{a + 1}}{b_{τ}} .

Each jump multiplies the running LCM by at least $p$ , so $0 < Y_{a} \leq \sum_{k \geq 1} p^{- k} = 1 / (p - 1)$ . Since every $r_{j} > p$ , $G_{τ} (1 / (p - 1)) \leq \sum_{j = 1}^{v} p^{- j} + p^{- v} / (p - 1) = 1 / (p - 1)$ , and $G_{τ} (0) > 0$ .

(iii) If $t \leq p^{a}$ , each exponent in $H_{P} (t)$ is at most the corresponding exponent in $P_{a}$ , so $H_{P} (t) ∣ P_{a}$ . Hence $P_{a} D_{P} - Y_{a} = P_{a} (1 + \sum_{t_{k} \leq p^{a}} H_{P} (t_{k})^{- 1})$ is an integer, and so is $K Y_{a} = P_{a} N - K (P_{a} D_{P} - Y_{a})$ . ◻

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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