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

Proof for an arbitrary finite prime set

Remark 5.1 (the sums over distinct running LCMs). For every finite set $P$ of primes with $| P | \geq 2$ , the sum $D_{P}$ is irrational.

Here we compare two tails rather than recover a word from one tail. Discreteness is used twice. Under rationality with denominator $K$ , the tails lie in $K^{- 1} Z$ ; a long common prefix makes two of them closer than $1 / K$ and hence equal. Next, corresponding blocks contain the same prime labels in possibly different orders. Their denominator products agree. Clearing that product leaves an integer difference of block numerators equal to a difference of tails in $(- 1, 1)$ . Both must vanish, and this conclusion propagates through the blocks.

We therefore need one rearranged block with a changed numerator. For two or three distinct primes, every change of order does this; for four or more, some do not. The torus construction isolates a pair of neighbouring jumps and reverses it. Other blocks may change too, but equality constrains every block separately, so one changed two-letter numerator suffices. We first establish the word identity, then construct the two arithmetic tails.

Words and tails.

The logarithmic flow.

Comparison of two tails.

Reversals near the origin.

Exact illustrations.

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.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

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