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

Smith factors of the selected staircase

The preceding argument only uses that a determinant is nonzero. An irrationality argument would also need control of integer coefficients after denominators are cleared. We examine that separate question for the selected minors. For integers $N \geq 1, r \geq 2$ , let $A_{N} (r)$ have entry $1$ on and below the diagonal and $r$ above it. This is the staircase used in Theorem 7.3. Its Smith factors determine the lattice index; they do not by themselves give a small integral linear form.

Remark 7.5 (Smith factors of the selected staircase). The Smith factors of $A_{N} (r)$ are $1, r - 1, \dots, r - 1$ , and $det A_{N} (r) = (1 - r)^{N - 1}$ .

The Smith calculation and its consequence for integral coefficients are ordinary arguments here; the existing Lean result proves nonvanishing of the selected minors, not these arithmetic statements.

Proof. Working from bottom to top, subtract row $i - 1$ from row $i$ for every $i > 1$ . Row $i$ then has just $1 - r$ in column $i$ . The first row is $(1, r, \dots, r)$ ; subtract $r$ times its first column from each later column. The resulting diagonal is $(1, 1 - r, \dots, 1 - r)$ . These are unimodular integral operations, and $1$ divides $r - 1$ , so this is its Smith form up to signs. Its determinant has the displayed sign. ◻

For the selected ${2, 3, 5}$ minor, the positive integers $U_{i}, V_{j}$ give

K = diag (U_{i})^{- 1} A_{N} (5) diag (5 V_{j})^{- 1} .

Consequently

det K = \frac{(- 4)^{N - 1}}{W}, W = 5^{N} \prod_{i} U_{i} \prod_{j} V_{j} .

More generally, an entrywise-integral diagonal clearing $diag (s) K diag (t)$ has diagonal entries $z_{i} = s_{i} t_{i} / (5 U_{i} V_{i}) \in Z ∖ {0}$ and determinant $(- 4)^{N - 1} \prod_{i} z_{i}$ . Its absolute determinant is therefore at least $4^{N - 1}$ , attained by $s_{i} = 5 U_{i}$ , $t_{j} = V_{j}$ . This optimality is restricted to diagonal entrywise clearings.

For the selected carry pattern, the injective nonnegative index families $I, J$ satisfy $v_{2} (W) \geq \sum_{i} I (i) + \sum_{j} J (j) \geq N (N - 1) \geq 2 (N - 1)$ . Thus $4^{N - 1}$ divides $W$ , and the reduced determinant is $\pm 1 / D$ , where $D = W / 4^{N - 1} \in N_{> 0}$ . Thus the factor $4^{N - 1}$ in the integer determinant is absorbed entirely by its denominator. For a linear form in one target value, a small rational prefactor likewise disappears when the coefficients are reduced to coprime integers. Indeed, for $K \in {GL}_{N} (Q)$ and rational $u, v$ , the Schur complement gives

det (\begin{matrix} K & u \\ v^{T} & X \end{matrix}) = det K (X - v^{T} K^{- 1} u) .

Write $v^{T} K^{- 1} u = s / t$ in lowest terms, with $t > 0$ . If a nonzero rational scalar multiple of $det K (X - s / t)$ has integer coefficients $a X + b$ , then $b = - a s / t$ . Since $gcd (s, t) = 1$ , we have $t ∣ a$ . Dividing the two coefficients by their common factor therefore leaves $\pm (t X - s)$ . This is the primitive integral coefficient polynomial, regardless of the smallness of $det K$ or a compatible rational change of basis. The conclusion concerns this construction with one target and the diagonal clearings above. It does not exclude new recurrences, multiple target values or stronger approximation data, and it does not settle the repeated running-LCM problem. Smith form is used here in its standard sense [14].

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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