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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How fast the product of the bases grows

The growth scale in the preceding criterion comes from the running LCM formula. The elementary estimate $W_{ℓ, h} \geq 2^{h}$ would suffice for bounds $G (B, a) = o (2^{a})$ with $B$ fixed; the exact formula gives $W_{ℓ, h} ≍ 8^{h}$ uniformly in $ℓ$ and allows $G (B, a) = o (8^{a})$ . For an irrational $S$ , the preceding lower-limit condition also covers dominating bounds outside these two classes. The same growth estimate will give a necessary lower bound on successful segment lengths.

Proposition 10.4 (growth of the product of consecutive bases). Put $θ_{3} = \log_{3} 2$ and $θ_{5} = \log_{5} 2$ . For all $ℓ \geq 0$ and $h \geq 1$ ,

W_{ℓ, h} = 2^{h} 3^{⌊ (ℓ + h) θ_{3} ⌋ - ⌊ ℓ θ_{3} ⌋} 5^{⌊ (ℓ + h) θ_{5} ⌋ - ⌊ ℓ θ_{5} ⌋}, \frac{8^{h}}{15} < W_{ℓ, h} < 15 \cdot 8^{h} .

Proof. Telescoping (14) gives $W_{ℓ, h} = P_{ℓ + h} / P_{ℓ}$ , and the displayed formula is that quotient written out. Each floor difference differs from $h θ_{p}$ by less than one, and $3^{θ_{3}} = 5^{θ_{5}} = 2$ , so the $3$ -factor lies strictly between $2^{h} / 3$ and $3 \cdot 2^{h}$ and the $5$ -factor strictly between $2^{h} / 5$ and $5 \cdot 2^{h}$ . Multiplying the three ranges gives the bounds. ◻

Corollary 10.5 (a fixed maximum length cannot cover arbitrarily late starts). Fix $B \geq 1$ coprime to $30$ and $H \geq 1$ . Only finitely many starts $ℓ$ admit a segment of length at most $H$ satisfying the residue inequality against $K$ .

Proof. A least positive residue never exceeds its modulus, so the residue inequality at $(ℓ, h)$ requires $K (B, ℓ + h) < W_{ℓ, h} < 15 \cdot 8^{h} \leq 15 \cdot 8^{H}$ . On the other hand $j_{a} \geq a - 1$ , since the powers $2, \dots, 2^{a - 1}$ already lie below $2^{a}$ , so $K (B, ℓ + h) \geq ⌊ B ((ℓ - 1)^{2} + 10 (ℓ - 1) + 27) / 9 ⌋$ , which tends to infinity with $ℓ$ . ◻

Corollary 10.5 is the exact reason a finite scan cannot approach the cofinal quantifier by widening its denominator range alone. For $ℓ \geq 1$ , the inequality $K (B, ℓ + h) > B ℓ^{2} / 9$ shows that any segment satisfying the residue inequality must also satisfy

3 h > \log_{2} K (B, ℓ + h) - \log_{2} 15 > \log_{2} B + 2 \log_{2} ℓ - \log_{2} 135.

This is only a necessary lower bound, not an asymptotic formula or an upper bound for the first successful length. An explicit threshold beyond which the residue inequality holds can be obtained from a positive lower bound for $δ = ⌈ B X_{ℓ} ⌉ - B X_{ℓ}$ and for $1 - δ$ , together with the displayed tail and product estimates. No such uniform information about the tails of $S$ is proved here.

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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