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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Uniform approximation of the rescaled kernel

Exact rank does not decide approximation. For the rescaled kernel, infinitely many columns remain a fixed distance apart, giving a uniform lower bound against every finite rank. For the original kernel, decay instead permits summable truncation. We prove these statements for the two different matrices. Write

\begin{matrix} (11) & C (i, j) = t^{⌊ x_{i} + y_{j} ⌋}, x_{i} = {i \log_{r} p}, y_{j} = {j \log_{r} q}, t = r^{- 1} . \end{matrix}

This is the remaining factor in (10) after removing the row and column factors. A matrix has finite separated rank if $A (i, j) = \sum_{ℓ < d} f_{ℓ} (i) g_{ℓ} (j)$ for some finite $d$ . Equivalently, its columns span a finite-dimensional space of real sequences. No continuity or boundedness of $f_{ℓ}, g_{ℓ}$ is assumed.

Theorem 7.6 (distance from matrices of finite separated rank). Let $p, q, r$ be pairwise distinct primes and let $C$ be as in (11). Then

inf_{A} sup_{i, j \geq 0} | C (i, j) - A (i, j) | = \frac{1 - t}{2} = \frac{r - 1}{2 r},

the infimum being over all matrices $A$ of finite separated rank, and it is attained by the constant matrix of value $(1 + t) / 2$ .

Proof. We first show that any two distinct columns of $C$ have supremum distance $1 - t$ . Every entry belongs to ${1, t}$ , with $C (i, j) = t$ precisely when $x_{i} + y_{j} \geq 1$ . Given $j \neq k$ , irrationality of $\log_{r} q$ makes $y_{j} \neq y_{k}$ , so suppose $y_{j} < y_{k}$ . By density of $(x_{i})$ we can choose $i$ with $1 - y_{k} < x_{i} < 1 - y_{j}$ ; the two entries in this row are $t$ and $1$ .

Suppose a finite-rank matrix $A$ approximated $C$ with error $ε = sup_{i, j} | C (i, j) - A (i, j) | < (1 - t) / 2$ . Its columns $A_{j}$ would be bounded sequences with $‖ A_{j} ‖_{\infty} \leq 1 + ε$ , and their span $V$ would be finite-dimensional. Although no boundedness was assumed for the separated factors, the supremum norm is defined on this span because each column is bounded. For distinct columns the triangle inequality gives

‖ A_{j} - A_{k} ‖_{\infty} \geq ‖ C_{j} - C_{k} ‖_{\infty} - 2 ε = 1 - t - 2 ε > 0.

Thus a bounded subset of $V$ would contain infinitely many points at a fixed positive distance from one another, contrary to total boundedness in finite dimension. This proves the lower bound.

For equality, take the constant matrix $A (i, j) = (1 + t) / 2$ . It has rank one, and its distance from every entry of $C$ is $(1 - t) / 2$ . ◻

The lower bound concerns the entire normalised matrix, not each finite restriction. For example, at $t = 1 / 5$ the matrices

T = (\begin{matrix} 1 / 5 & 1 \\ 1 / 5 & 1 / 5 \end{matrix}), A = (\begin{matrix} 3 / 10 & 9 / 10 \\ 1 / 10 & 3 / 10 \end{matrix})

satisfy $det A = 0$ and $‖ T - A ‖_{max} = 1 / 10 < 2 / 5$ . Here $‖ \cdot ‖_{max}$ is the largest absolute entry. This finite example has no bearing on the infinite family of separated columns used in the proof.

For the original kernel, the decaying row and column factors instead allow approximation in the summation norm. Let $K^{(N)} (i, j, k) = K (i, j, k)$ for $i < N$ and $K^{(N)} (i, j, k) = 0$ otherwise. This is a sum of at most $N$ terms separated between $i$ and $(j, k)$ . Since $H (x) > x^{3} / (p q r)$ for $x \geq 1$ , geometric summation gives

\begin{matrix} (12) & \sum_{i, j, k \geq 0} | K (i, j, k) - K^{(N)} (i, j, k) | \leq \frac{p q r p^{- 3 N}}{(1 - p^{- 3}) (1 - q^{- 3}) (1 - r^{- 3})} . \end{matrix}

The same bound controls the supremum of the entrywise errors, so these finite separated-rank approximants converge both in $ℓ^{1}$ and uniformly to the original kernel. The positive uniform lower bound in Theorem 7.6 belongs to the rescaled matrix. Diagonal rescaling preserves exact rank, but the rescaling factors here are unbounded and do not preserve uniform error estimates. Neither the rank obstruction nor these approximation bounds decide whether the scalar sum is rational.

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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