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

Function-theoretic representations

The elementary proof of irrationality of $D_{{2, 3, 5}}$ gives no transcendence conclusion. Recall that

D_{{2, 3, 5}} = 1 + \sum_{t \in {2^{n}, 3^{n}, 5^{n} : n \geq 1}} \frac{1}{2^{⌊ \log_{2} t ⌋} 3^{⌊ \log_{3} t ⌋} 5^{⌊ \log_{5} t ⌋}} .

Each running LCM is counted once at its first jump, unlike the repeated sum. The bases in (14) come from these same jumps. For two primes a representation by one Hecke–Mahler value proves transcendence (Section 4). A corresponding three-prime representation has not been constructed here.

Problem 12.7 (an exact function representing the distinct-height sum). Transcendence of $D_{{2, 3, 5}}$ is not proved here. For a functional approach to it, first specify the function, its coefficient field and convergence domain, then prove an exact identity for $D_{{2, 3, 5}}$ , and only then apply a stated value theorem. A conditional theorem must display the extra nondegeneracy assumption; a no-representation theorem must specify the class it excludes.

For $(p, q, r) = (2, 3, 5)$ , the two slopes in the rank proof are $\log_{5} 2 = θ_{5}$ and $\log_{5} 3 = θ_{5} / θ_{3}$ , not $θ_{3}$ and $θ_{5}$ . Each is irrational: a rational value would equate a positive power of $5$ with a positive power of $2$ or $3$ . The proof chooses the row and column indices independently. It therefore needs no rational independence of $1, θ_{3}, θ_{5}$ and no density assertion for the singly indexed orbit $({n θ_{3}}, {n θ_{5}})$ . Any functional approach that needs such a stronger hypothesis must establish it separately.

A finite-dimensional encoding is useful here only if its data determine the represented function: equal encodings must give equal functions. The Lean sources impose this through a factorisation of the linear map to functions through a finite-dimensional space. The resulting space of functions is then finite-dimensional (identity for the carry, map to functions, finite-dimensional conclusion). They also bound the carry residue and its digit in their stated intervals (residue bound, digit bound). These conditional linear-algebra statements neither construct a function representing $D_{{2, 3, 5}}$ nor show that a corresponding function space must be infinite-dimensional.

The cited value theorems and their hypotheses.

Pellarin’s rank-one theorem concerns quadratic irrational slopes and algebraic evaluation points in the convergence domain [31]. The prime-logarithm slopes used here are not quadratic: each $\log_{r} p$ is irrational by unique factorisation, and the Gelfond–Schneider theorem then makes it transcendental, since $r^{\log_{r} p} = p$ is algebraic.¹ This rules out direct substitution of those slopes into Pellarin’s quadratic-slope theorem. It does not rule out a different representation of the scalar sum, and it does not affect the two-prime argument above, which uses a theorem for general irrational slopes.

In Pellarin’s reduced-slope normalisation, the rank-one condition says that the evaluation points arise from a common point by the quadratic order’s monomial action, up to multiplication by torsion points. For a single evaluation point it is automatic: use that point itself and the identity action. For several points it is a condition to check, not a consequence of having two lattice coordinates. Under it, algebraic independence of the values is equivalent to $Q$ -linear independence of the specified auxiliary formal Laurent series; the rank-one condition alone does not imply independence. This module rank is not the rank of our reciprocal kernel. No exact representation suitable for this theorem has been constructed here, but infinite kernel rank alone does not preclude one.

For one variable, Adamczewski and Faverjon’s proofs of Nishioka’s theorem and the lifting theorem are regular-point results [32]. Regularity means that the system matrix stays defined and invertible along the evaluation orbit. Their multivariate lifting theorem [33] additionally assumes an admissible transformation–point pair. Its growth, decay and nonvanishing conditions are specified in Definition 5.1; none follows merely from writing a double sum. Lifting transfers a relation among values to one among functions; a transcendence application must exclude the latter. A Mahler system alone cannot do this, as the constant function $1$ illustrates.

Regularity of a preselected system is not, however, a universal prerequisite for Mahler’s method. The same paper’s Theorem 1.1 treats univariate Mahler functions at algebraic points in the punctured unit disc where their values are defined, without that regularity hypothesis. The hypotheses must therefore be matched to the particular theorem being invoked. No exact Mahler representation or suitable evaluation data for this sum over distinct running LCMs are constructed here. The locators for this paper refer to its 68-page author manuscript, not the older 52-page arXiv version.

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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