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.

Start here, by the editorial route

In this paper

The remaining arithmetic questions

The finite recursions are explicit; the missing step is to prove that successful segments exist with the required unbounded quantifiers. Weighted shifts preserve the value of the series, but preservation does not supply cancellation. We calculate the surviving boundary terms, then identify the extra hypotheses needed by integer-recurrence and function-theoretic approaches. The irrationality of $S$ remains open here.

An exact weighted-shift identity for the repeated series

Boundary terms in weighted finite differences

Function-theoretic representations

Hypotheses for auxiliary integer recurrences

The unresolved repeated-sum question

Verification and reproducibility

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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