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

Words obtained from a linear flow

We apply the interchange argument to the order in which finitely many linear functions take integer values. Their slopes will be assumed linearly independent over $Q$ ; this hypothesis is not assumed for reciprocal prime logarithms. Interchanging two late crossings then determines the rational relations among the corresponding subseries, without requiring recovery of every letter. The arguments in this section remain ordinary proofs awaiting formalisation.

Let $A$ be a finite alphabet. Give $a \in A$ an integer radix $b_{a} \geq 2$ and a rational coefficient $c_{a}$ . For $w = w_{1} w_{2} \dots$ , define

Q_{0} = 1, Q_{n} = \prod_{j = 1}^{n} b_{w_{j}}, V_{c} (w) = \sum_{n \geq 1} \frac{c_{w_{n}}}{Q_{n}}, T_{a} (x) = \frac{c_{a} + x}{b_{a}} .

Write $σ w = w_{2} w_{3} \dots$ and $X_{w} = \overset{―}{{σ^{n} w : n \geq 0}}$ in the one-sided product topology. A word belongs to $X_{w}$ precisely when each of its finite prefixes begins some shifted word. It need not itself be a shift. The first letter acts by the outermost map: if $u = u_{1} \dots u_{m}$ , then $V_{c} (u z) = T_{u_{1}} \circ \dots \circ T_{u_{m}} (V_{c} (z))$ . Put $B (u) = \prod_{j \leq m} b_{u_{j}}$ , with $B (\emptyset) = 1$ .

Remark 6.1 (interchanging two letters after a long common prefix). Suppose $V_{c} (w)$ is rational. If for distinct letters $a, b$ and arbitrarily long words $u$ there is a common infinite suffix $z$ such that both $u a b z$ and $u b a z$ belong to $X_{w}$ , then

c_{a} (b_{b} - 1) = c_{b} (b_{a} - 1) .

If such interchanges join every pair of letters along a connected graph, $V_{c} (w)$ is rational precisely when there is $λ \in Q$ with $c_{a} = λ (b_{a} - 1)$ for all $a$ ; in that case $V_{c} (w) = λ$ .

Proof. First clear the common denominator of the coefficients, so that all $c_{a}$ are integers. The recurrence $V_{c} (σ w) = b_{w_{1}} V_{c} (w) - c_{w_{1}}$ shows that, if $V_{c} (w) = p / K$ , then every shifted value belongs to $K^{- 1} Z$ . For $λ_{a} = c_{a} / (b_{a} - 1)$ , the positive weights

\frac{b_{w_{n}} - 1}{Q_{n}} = \frac{1}{Q_{n - 1}} - \frac{1}{Q_{n}}

sum to one. Hence $V_{c} (v)$ lies between $min_{a} λ_{a}$ and $max_{a} λ_{a}$ for every word $v$ . Sharing $m$ initial letters changes a value by at most the width of this interval divided by $B (u)$ . Thus $V_{c}$ is continuous. Since $K^{- 1} Z$ is closed, the lattice condition passes from the actual shifts to all of $X_{w}$ .

The order of the maps matters. A direct calculation gives, for every $x$ ,

T_{a} (T_{b} (x)) - T_{b} (T_{a} (x)) = \frac{c_{a} (b_{b} - 1) - c_{b} (b_{a} - 1)}{b_{a} b_{b}} .

The values of $u a b z$ and $u b a z$ therefore differ by this number divided by $B (u)$ . Their difference belongs to $K^{- 1} Z$ . If its numerator were nonzero, its absolute value would be positive but smaller than $1 / K$ for a sufficiently long $u$ , a contradiction. On a connected interchange graph all $λ_{a}$ agree. Conversely, for $c_{a} = λ (b_{a} - 1)$ the displayed positive weights telescope and give $V_{c} (v) = λ$ for every $v$ . Dividing by the earlier common denominator restores the statement for rational coefficients. ◻

The common suffix is essential: distinct words can have the same affine map. For example, the blocks $(5, 7, 3, 2)$ and $(7, 2, 3, 5)$ have the same map $(51 + x) / 210$ . Choosing the second block at block indices $1, 2, 4, 8, \dots$ produces a non-eventually-periodic word with rational value $51 / 209$ . Thus aperiodicity and bounded tails do not suffice, even with integer bases; the interchange hypothesis is essential to the argument. Nor does the lemma assert sufficiency when its graph is disconnected.

Now let $r_{1}, \dots, r_{d} > 0$ , $d \geq 2$ , be linearly independent over $Q$ . Record the letter $i$ at each positive time $t$ for which $r_{i} t + θ_{i} \in Z$ , where $θ_{i} \in [0, 1)$ . Assume that no two of these functions take integer values simultaneously at positive time. In chronological order, the crossings give an infinite word $w$ on ${1, \dots, d}$ .

Remark 6.2 (isolated interchanges in the crossing sequence). Every pair of distinct labels satisfies the late-interchange hypothesis of Remark 6.1 for $X_{w}$ .

Proof. Fix a label $a$ . Immediately after its sufficiently late crossings, the remaining coordinates of the flow are

θ_{i} - \frac{r_{i}}{r_{a}} θ_{a} + k \frac{r_{i}}{r_{a}} (\mod 1) (i \neq a),

where $k$ runs through all sufficiently large integers. These points are dense in the $(d - 1)$ -torus. Indeed, $1$ and the $r_{i} / r_{a}$ for $i \neq a$ are rationally independent. For every nonzero integral character of the torus, its average along this orbit is a geometric sum divided by its length and tends to zero. Trigonometric approximation of continuous functions then shows that every nonempty open set is visited infinitely often. This also proves density after any finite initial segment is discarded.

Fix $b \neq a$ and an arbitrarily large integer $L$ , and work on the section just after an $a$ -crossing. Set $T = L / r_{a}$ and choose the phase of $b$ so that the $b$ -coordinate crosses at $T$ . This phase is nonzero, since $L r_{b} / r_{a} \notin Z$ . Choose the remaining phases outside the countable sets that cause a crossing at zero, a third crossing at $T$ , or any other positive simultaneous crossing. The resulting singular configuration has exactly the one double crossing of $a, b$ at $T$ : a second meeting of these two coordinates would make $r_{b} / r_{a}$ rational.

Perturb the $b$ phase on opposite sides. On each bounded time interval, a sufficiently small perturbation changes only this pair’s order. Taking one-sided limits gives $u a b z$ and $u b a z$ with the same infinite suffix $z$ . No fixed nonzero perturbation is required to preserve all later crossings: agreement on each finite interval suffices for these limiting words. Their common prefix has at least $L - 1$ letters, from the earlier $a$ -crossings. For each finite prefix of either limit, a small open set of nonsingular section phases realises that prefix. The dense actual $a$ -section orbit meets that set, so the prefix occurs at the beginning of an actual shifted word. Consequently both limiting words belong to $X_{w}$ . Letting $L$ grow proves the assertion. ◻

Remark 6.3 (rational relations among the subseries). Under the preceding hypotheses on the linear flow, choose integers $b_{i} \geq 2$ , put $Q_{n} = \prod_{k \leq n} b_{w_{k}}$ and

S_{i} = \sum_{n : w_{n} = i} \frac{1}{Q_{n}} (1 \leq i \leq d) .

For every $c_{i} \in Q$ ,

\sum_{i = 1}^{d} c_{i} S_{i} \in Q ⟺ c_{i} = λ (b_{i} - 1) for all i and some λ \in Q .

The rational value is $λ$ . Equivalently, all rational relations $a_{0} + \sum_{i} a_{i} S_{i} = 0$ are multiples of $(- 1, b_{1} - 1, \dots, b_{d} - 1)$ . Thus the span of $1, S_{1}, \dots, S_{d}$ has dimension $d$ ; each $S_{i}$ and each nonempty proper unweighted sub-sum is irrational. The full unweighted sum is irrational exactly when the radices are not all equal.

The following argument proves the assertion of the remark in ordinary mathematics; formalisation and external peer review remain outstanding.

Proof. Absolute convergence gives $\sum_{i} c_{i} S_{i} = V_{c} (w)$ . The previous two remarks make the interchange graph complete, so they give both directions and the stated value. The relation space follows by moving the rational value to the constant coordinate. An indicator vector of a nonempty proper set has both zero and one entries and cannot be a multiple of the strictly positive vector $(b_{i} - 1)_{i}$ . The all-ones vector is such a multiple precisely when all the $b_{i}$ agree. ◻

For example, take

(r_{1}, r_{2}, r_{3}, r_{4}) = (1, \sqrt{2}, \sqrt{3}, \sqrt{6}), (b_{1}, b_{2}, b_{3}, b_{4}) = (2, 3, 5, 7), θ_{i} = 0.

The four rates form the standard basis of the biquadratic field $Q (\sqrt{2}, \sqrt{3})$ over $Q$ ; the ratio of any two distinct rates is irrational, so positive crossings never coincide. Write $(d_{1}, d_{2}, d_{3}, d_{4}) = (1, 2, 3, 6)$ . Counting crossings up to the $n$ th crossing of the $i$ -th coordinate gives the explicit series

\begin{matrix} (9) & S_{i} = \sum_{n \geq 1} {(\prod_{j = 1}^{4} b_{j}^{⌊ n \sqrt{d_{j} / d_{i}} ⌋})}^{- 1} . \end{matrix}

The only rational relation among $1, S_{1}, S_{2}, S_{3}, S_{4}$ is generated by $S_{1} + 2 S_{2} + 4 S_{3} + 6 S_{4} = 1$ . In particular their rational span has dimension four and $\sum_{i} S_{i}$ is irrational. This does not provide an irrationality measure or transcendence result. In the related two-letter Hecke–Mahler setting, stronger transcendence results exist [15]; the example’s four independent rates do not satisfy that two-letter hypothesis. The ordinary two-wall argument for the distinct running-LCM sum avoids this assumption on reciprocal prime logarithms. The repeated running-LCM question remains open.

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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