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

Worked finite examples

At the first ten integer cutoffs, the running LCM changes by a factor $2$ at $2, 4, 8$ , by $3$ at $3, 9$ , and by $5$ at $5$ . The table in Section 3 also shows constant values on ${5, 6, 7}$ and ${9, 10}$ , as in Proposition 3.2. In particular, $L (10) = 8 \cdot 9 \cdot 5 = 360$ is the LCM of $1, 2, 3, 4, 5, 6, 8, 9, 10$ , the supported integers at most $10$ .

For Proposition 3.4 take $(p, q, r) = (2, 3, 5)$ and the box $B (1, 1, 1)$ , whose eight points carry the smooth values $1, 2, 3, 5, 6, 10, 15, 30$ and the running LCMs

\begin{array}{ccccccccc} p^{i} q^{j} r^{k} & 1 & 2 & 3 & 5 & 6 & 10 & 15 & 30 \\ H & 1 & 2 & 6 & 60 & 60 & 360 & 360 & 10800 \end{array}

Six running LCMs occur, two of them twice: the points $5$ and $6$ share the running LCM $60$ , and $10$ and $15$ share the running LCM $360$ . The identity reads

1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{60} + \frac{1}{60} + \frac{1}{360} + \frac{1}{360} + \frac{1}{10800} = 1 + \frac{1}{2} + \frac{1}{6} + \frac{2}{60} + \frac{2}{360} + \frac{1}{10800} = \frac{18421}{10800},

and the two coefficients $2$ carry the whole content of the regrouping on this box. Where the running LCMs are pairwise distinct the identity is a relabelling.

Multiplying block radices along a run of blocks gives the product of the prime multipliers at the jumps in that run: for instance $b_{1} b_{2} = 60$ is the product of the four multipliers at $3, 4, 5, 8$ . Block $4$ is the only one among the first six containing an internal power of each odd prime, whereas block $5$ contains neither, so its radix falls back to the terminal factor alone.

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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