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

The finite-scan histogram

For completeness we reproduce the first-success counts from the larger scan in Section 11. It tested $B \leq 5000$ coprime to $30$ and $100 \leq ℓ \leq 3000$ , stopping each search at depth $24$ . The saved output (scan-5000-100-3000.json) gives $3,869,934$ pairs, a segment satisfying the inequality in every case, and the following first-success histogram:

\begin{array}{crrrrrrrrrrrrrrr} h & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ # & 1 & 104 & 812 & 5437 & 51409 & 237423 & 735450 & 1431226 & 1132756 & 236752 & 34910 & 3076 & 521 & 49 & 8 \end{array}

The first case attaining the maximal observed length $18$ has $B = 917$ and $ℓ = 2980$ . The smaller run (scan-1000-100-500.json) over $B \leq 1000$ coprime to $30$ and $100 \leq ℓ \leq 500$ gives $106,666$ pairs and maximal first successful length $14$ , first attained at $B = 359$ and $ℓ = 291$ . The two cases have the following data.

	Larger scan	Smaller scan
Denominator $B$	$917$	$359$
Start $ℓ$	$2980$	$291$
Length $h$	$18$	$14$
Endpoint jump index	$6179$	$627$
Product of bases	$18139852800000000$	$5038848000000$
Accumulated numerator	$13196471407660025821045$	$25864575212865807$
Least positive residue	$76322101735$	$213175287$
Upper bound	$3896420420$	$15932659$

These histograms concern a bounded region. The product-growth law gives a necessary lower bound on a possible length of such a segment, not a distribution law for residues or first successful lengths. No generic-residue model is used as evidence for the residue inequality at arbitrarily late starts.

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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