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 and tails.

For an infinite word $u = u_{1} u_{2} \dots$ over $P$ , define $V (u) = \sum_{j \geq 1} (u_{1} \dots u_{j})^{- 1}$ . For a nonempty finite word $σ = σ_{1} \dots σ_{n}$ , put

Π (σ) = σ_{1} \dots σ_{n}, f (σ) = \sum_{j = 1}^{n} σ_{j + 1} \dots σ_{n} .

The final term in $f$ is $1$ . Thus $f$ is the numerator of the block contribution when its denominator is the product of the block letters. For two distinct letters $a, b$ , we have $f (a b) = b + 1$ and $f (b a) = a + 1$ . Reversing this pair changes $f$ by the nonzero integer $b - a$ . Removing a prefix gives

\begin{matrix} (8) & V (σ u) = \frac{f (σ) + V (u)}{Π (σ)} . \end{matrix}

We apply this identity to $x_{k} = V (q_{k + 1} q_{k + 2} \dots) = \sum_{j > k} Q_{k} / Q_{j}$ . Every suffix contains every prime of $P$ , so $0 < x_{k} < 1 / (min P - 1) \leq 1$ .

Remark 5.2 (integral tails). If $D_{P} = N / K$ with integers $N$ and $K \geq 1$ , then $K x_{k}$ is an integer for every $k \geq 1$ .

Proof. Each $Q_{i}$ with $i \leq k$ divides $Q_{k}$ . Multiplication of the defining series by $Q_{k}$ therefore gives

K x_{k} = Q_{k} N - K (Q_{k} + \sum_{i \leq k} \frac{Q_{k}}{Q_{i}}) \in Z .

◻

Two words with a long common prefix have close values. When the words differ only by rearrangements, equality of their values imposes a stronger condition on every rearranged block.

Remark 5.3 (rearranged blocks). Let $u = σ_{1} σ_{2} \dots$ and $u^{'} = σ_{1}^{'} σ_{2}^{'} \dots$ be infinite words, where each $σ_{i}$ is a nonempty finite word and $σ_{i}^{'}$ is a rearrangement of it, and suppose that every suffix of $u$ or $u^{'}$ starting at a block boundary has value in $(0, 1)$ .

If $V (u) = V (u^{'})$ , then $f (σ_{i}) = f (σ_{i}^{'})$ for every $i$ .
If $σ_{i} = σ_{i}^{'}$ for every $i < i_{0}$ , then $| V (u) - V (u^{'}) | < 1 / Π (σ_{1} \dots σ_{i_{0} - 1})$ .

Proof. Let $U_{i}$ and $U_{i}^{'}$ be the values starting at block $i$ . Since the two blocks have the same product, (8) yields

Π (σ_{i}) (U_{i} - U_{i}^{'}) = f (σ_{i}) - f (σ_{i}^{'}) + U_{i + 1} - U_{i + 1}^{'} .

If $U_{i} = U_{i}^{'}$ , the integer on the right equals a number in $(- 1, 1)$ , and hence both differences vanish. Induction proves (i). If the preceding blocks agree, repeated removal of their common prefix gives

V (u) - V (u^{'}) = \frac{U_{i_{0}} - U_{i_{0}}^{'}}{Π (σ_{1} \dots σ_{i_{0} - 1})},

which proves (ii). ◻

Remark 5.4 (short rearrangements). The map $f$ is injective on the orderings of any set of at most three distinct primes.

Proof. For two letters, $f (a b) = 1 + b$ . For three, $f (a b c) = 1 + c (1 + b)$ . Suppose two orderings of ${a, b, c}$ have the same value. If their last letters agree, their middle letters agree too. Otherwise we may write $a (1 + s) = b (1 + t)$ , with $s \in {b, c}$ and $t \in {a, c}$ . The choices $(s, t) = (b, a)$ and $(c, c)$ imply $a = b$ ; the other two imply $b ∣ a$ or $a ∣ b$ . Each is impossible for distinct primes. ◻

There is a collision on four letters: $f (5, 7, 3, 2) = f (7, 2, 3, 5) = 51$ . Consequently, a change in order alone will not suffice for larger $P$ . We shall arrange an isolated interchange of two letters, for which $f (a b) - f (b a) = b - a \neq 0$ .

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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