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

Repeated tails and prime-power subseries

We compare the two normalisations before considering the sums over pure prime powers. The subseries assertions in this subsection have ordinary proofs and remain unformalised. For $P = {2, 3, 5}$ the two tail recurrences have the same bases, $b_{τ_{a}} = P_{a + 1} / P_{a} = b_{a}$ , but their coefficients behave differently:

Y_{a + 1} = b_{a} Y_{a} - μ_{τ_{a}}, X_{a + 1} = b_{a} X_{a} - m_{a} .

The first identity is Remark 5.6(ii); the second is Theorem 8.2, with the notation of Section 8. The five types give only the coefficients $1, 3, 9, 13$ , whereas the weighted block counts satisfy $m_{a} = Θ ((a + 1)^{2})$ by Lemma 8.3. Consequently $X_{a} > m_{a} / 30$ is unbounded. Under rationality the integral values $B X_{a}$ need not repeat, and the maps $y \mapsto (m_{a} + y) / b_{a}$ form an infinite family depending on $a$ . Even equality of two tails would not identify their later coefficients. The residue criterion in Section 10 compares an endpoint residue with a quadratic bound instead. The example after Proposition 13.2 shows the limit of an argument using just bases and growth: the same bases admit quadratic numerators ${\hat{m}}_{a}$ with $\sum_{a} {\hat{m}}_{a} / P_{a + 1} = 9$ .

The two-tail proof of Remark 5.1 stops at the same place. Over the prime powers the repeated sum is $R_{P} = \sum_{k \geq 0} c_{k} / Q_{k}$ , where $c_{k}$ counts the smooth integers in $[t_{k}, t_{k + 1})$ , with $t_{0} = 1$ . The two tails after $t_{k}$ and after $t_{k} p^{n}$ still have words that differ by rearranging chains, and a rational value still makes the scaled tails integers of one denominator. But the scaled tails are unbounded, as the $X_{a}$ are, and Remark 5.3 needs every tail in $(0, 1)$ : the integer $f (σ_{i}) - f (σ_{i}^{'})$ is shown to vanish because it equals a difference of two tails. With unbounded tails that step fails, the digits $c_{k}$ move with the rearrangement as well, and equality of two tails no longer forces agreement on each rearranged run. The irrationality of $R_{P}$ for $| P | \geq 3$ remains open.

Prime-power subseries.

The comparison of two tails applies to each prime-power subseries. For $p \in P$ put

E_{p} = \sum_{α \geq 0} \frac{1}{L (p^{α})} = \sum_{α \geq 0} \frac{1}{H_{P} (p^{α})},

the sub-sum of $R_{P}$ over the terms at which the smooth integer is a power of $p$ . For $P = {p, q}$ with $p < q$ these are the numbers $A = E_{p}$ and $1 + B_{*} = E_{q}$ of the proof of Theorem 1.2.

Remark 5.9 (sub-sums and the catalogue sum). Let $P$ be a finite set of primes with $| P | \geq 2$ .

$D_{P} = \sum_{q \in P} E_{q} - (| P | - 1)$ .
If $P = {p, q}$ , then $H_{P} (p^{i} q^{j}) = H_{P} (p^{i}) H_{P} (q^{j})$ for all $i, j \geq 0$ , and $R_{P} = E_{p} E_{q}$ .
If $P = {p, q, r}$ , then $H_{P} (p^{i} q^{j} r^{k}) = H_{P} (p^{i}) H_{P} (q^{j}) H_{P} (r^{k}) p^{κ_{p} (j, k)} q^{κ_{q} (i, k)} r^{κ_{r} (i, j)}$ , where $κ_{p} (j, k) = ⌊ j \log_{p} q + k \log_{p} r ⌋ - ⌊ j \log_{p} q ⌋ - ⌊ k \log_{p} r ⌋ \in {0, 1}$ and $κ_{q}, κ_{r}$ are defined in the same way. For a general finite $P$ the corresponding exponents lie in ${0, \dots, | P | - 2}$ .

Proof. (i) Each term $1 / Q_{k}$ of $D_{P}$ with $k \geq 1$ is a term of $E_{q_{k}}$ and of no other $E_{q}$ , and each $E_{q}$ also contains the term $1$ . (ii) and (iii) The exponent of $p$ in $H_{P} (\prod_{q \in P} q^{e_{q}})$ is $e_{p} + ⌊ \sum_{q \neq p} e_{q} \log_{p} q ⌋$ , and its exponent in $\prod_{q \in P} H_{P} (q^{e_{q}})$ is $e_{p} + \sum_{q \neq p} ⌊ e_{q} \log_{p} q ⌋$ . For $m$ nonnegative reals the floor of the sum exceeds the sum of the floors by an integer in ${0, \dots, m - 1}$ . With two primes $m = 1$ , so the running LCMs factor, and summing $H_{P} (p^{i})^{- 1} H_{P} (q^{j})^{- 1}$ over $i, j \geq 0$ gives $E_{p} E_{q}$ . ◻

With three primes $R_{P}$ is therefore the product of the three single-prime sums twisted by the carries $κ$ , and the plain product $E_{p} E_{q} E_{r}$ differs from it.

Remark 5.10 (the single-prime sub-sums). Let $P$ be a finite set of primes with $| P | \geq 2$ and let $p \in P$ . Then $E_{p}$ is irrational.

For $P = {p, q}$ , $E_{p}$ is $A$ or $1 + B_{*}$ in the proof of Theorem 1.2, an affine function of the Hecke–Mahler value $F_{θ} (1 / p, 1 / q)$ with nonzero slope; for $p < q$ , $E_{p} = p / (p - 1) - (q - 1) F_{θ} (1 / p, 1 / q)$ by (6). It is transcendental by the theorem of Bugeaud and Laurent [15], in the case due to Loxton and van der Poorten [16]. We cite these sources for this case and have not re-verified its earlier attribution history.

Proof. We modify the word argument so that it counts only occurrences of $p$ . For an infinite word $u$ and a finite word $σ = σ_{1} \dots σ_{n}$ , put

V_{p} (u) = \sum_{i : u_{i} = p} \frac{1}{u_{1} \dots u_{i}}, f_{p} (σ) = \sum_{j : σ_{j} = p} σ_{j + 1} \dots σ_{n} .

Then $V_{p} (σ u) = (f_{p} (σ) + V_{p} (u)) / Π (σ)$ . Writing $y_{k} = V_{p} (q_{k + 1} q_{k + 2} \dots)$ , we have

E_{p} = 1 + \sum_{k : q_{k} = p} \frac{1}{Q_{k}}, Q_{k} E_{p} = Q_{k} + \sum_{j \leq k, q_{j} = p} \frac{Q_{k}}{Q_{j}} + y_{k} .

In an actual suffix, the prefix product at the $j$ th occurrence of $p$ is at least $p^{j}$ , with strict inequality after another prime has occurred. Hence $0 < V_{p} (u) < 1 / (p - 1) \leq 1$ , and rationality $E_{p} = N / K$ would imply $K y_{k} \in Z$ . The proof of Remark 5.3 therefore applies to $V_{p}, f_{p}$ as well. On a chain of distinct primes containing $p$ , only one term remains in $f_{p}$ : the product of the primes following $p$ . Unique factorisation detects a change in that set, without recovering its order. This weaker requirement avoids the collisions of the full block numerator $f$ .

For each $q \neq p$ , the form $θ_{p} - θ_{q}$ is nonzero on the rational subspace $L$ used in Section 5.1. We may therefore choose $v \in L$ with $| v_{q} | \leq 1$ for every $q$ and $v_{q} \neq v_{p}$ for $q \neq p$ , where $v_{q} = θ_{q} (v)$ , and put $μ = min_{q \neq p} | v_{p} - v_{q} | > 0$ . Suppose $E_{p} = N / K$ . Choose a jump $t_{k}$ labelled $p$ , and choose $T, ν, g, β$ as in the general irrationality proof, with $2^{ν} \geq K$ . Set $η = 1 / 10$ and first fix

0 < ε < min {β, g / 2, \log 2 / (2 | P |), η / 4, η μ / 4} .

Dirichlet’s theorem then supplies arbitrarily large $n$ and integers $m_{q} \geq 1$ such that $δ_{p} = 0$ and $0 < | δ_{q} | = | n \log p - m_{q} \log q | < ε$ for $q \neq p$ . We compare $t_{k}$ with $t_{m} = t_{k} p^{n}$ and cut crossings at gaps greater than $2 ε$ . As before, each chain consists of distinct primes and is merely rearranged by the phase shift. The common prefix of $ν$ letters and the integral-tail argument give $y_{k} = y_{m}$ , and hence $f_{p} (C) = f_{p} (C^{'})$ for every chain and its rearrangement.

We shall contradict this equality near the origin of $T_{0}$ . Put

\begin{aligned} b_{0} & = min_{q \neq p} \frac{| δ_{q} |}{| v_{p} - v_{q} |}, & B & = {q \neq p : δ_{q} < 0}, \\ A_{+} & = {q \neq p : v_{q} < v_{p}}, & A_{-} & = {q \neq p : v_{q} > v_{p}} . \end{aligned}

Since $0 < b_{0} \leq ε / μ < η / 4$ and $A_{+}, A_{-}$ partition the nonempty set $P ∖ {p}$ , one of these two sets, say $A_{s}$ with $s \in {+ 1, - 1}$ , differs from $B$ . Take $b = s b_{0} / 2$ and consider $z = b v + Θ (- η / 2)$ . For each $q$ , its crossing nearest $η / 2$ occurs at

τ_{q} = η / 2 - b v_{q} \in (3 η / 8, 5 η / 8), τ_{q} - τ_{p} = b (v_{p} - v_{q}) .

Every other $q$ -crossing lies outside $(- η / 4, 5 η / 4)$ , and for $q \neq p$ the displayed difference is nonzero with absolute value at most $| δ_{q} | / 2$ . We choose a neighbourhood $U$ of $z$ in $T_{0}$ on which there are no crossings in $[- η / 4, η / 4] \cup [3 η / 4, 5 η / 4]$ , each prime has a crossing in $(η / 4, 3 η / 4)$ , and every difference $τ_{q} - τ_{p}$ retains its sign and has absolute value less than $| δ_{q} |$ . These conditions are open and hold strictly at $z$ .

By density, $θ_{k} + Θ (t^{*}) \in U$ for some $t^{*} > T + 2$ . The resulting crossings near $t^{*}$ form one chain: each is within $ε$ of the $p$ -crossing and every other crossing is more than $η / 2 > 2 ε$ away. In this chain the followers of $p$ form $A_{s}$ . After the shift, $q$ follows $p$ precisely when $τ_{q} - τ_{p} > δ_{q}$ , which, by the bound $| τ_{q} - τ_{p} | < | δ_{q} |$ , is equivalent to $δ_{q} < 0$ . Its followers now form $B \neq A_{s}$ , contradicting $f_{p} (C) = f_{p} (C^{'})$ . ◻

This argument uses a change of followers and so needs no two-wall lemma. Either of $A_{+}, A_{-}$ may be empty, but they are distinct when $| P | \geq 2$ . For $P = {p}$ the sum is $p / (p - 1)$ , and this is precisely where the choice of two distinct follower sets fails. The strict conditions defining $U$ are needed to carry the construction to an actual orbit.

For $P = {2, 3, 5}$ , $p = 2$ , $t_{k} = 2^{20}$ and $n = 65$ , the first rearranged chain is $(2, 3)$ against $(3, 2)$ after $131$ letters, with $f_{2}$ equal to $3$ and $1$ , and the difference of the two tails, computed exactly to $300$ letters, has $\log_{10} | y_{k} - y_{m} | = - 58.098$ .

For the repeated sum $R_{P}$ , changing the scale from $t_{k}$ to $t_{k} p^{n}$ brings in early terms from the rays $p^{α} w$ , with $w$ supported on $P ∖ {p}$ . The comparison above accounts only for the ray $w = 1$ , so it gives no corresponding estimate for the difference of the full repeated tails. In the case $P = {2, 3, 5}$ those normalised tails are unbounded, as the lower bounds below show.

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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