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.

Start here, by the editorial route

In this paper

Finite computations and their limits

We first calculate individual residue certificates, then enclose the sum using twelve blocks to exclude small rational denominators. The larger scan and two larger exclusions are separate saved computations; their programs and outputs are in the large-computation directory. Each certificate excludes its stated finite family. None establishes the criterion for every denominator beyond every starting threshold.

Checking individual segments

We calculate each row using the integer-only integer-arithmetic checker. For block $a$ , enumerate the pairs with $3^{j} 5^{k} < 2^{a + 1}$ , choose the unique $i \geq 0$ for which $2^{i} 3^{j} 5^{k} \in [2^{a}, 2^{a + 1})$ , and sum $P_{a + 1} / (2 H (2^{i} 3^{j} 5^{k}))$ . The recursions for $W$ and $F$ then give the residue in each row. Powers and comparisons are integral throughout. The columns give the denominator $B$ , the start $ℓ$ , the length $h$ , the endpoint jump index $j_{ℓ + h}$ , the product $W$ of the bases, the accumulated numerator $F$ , the residue $R = {lpr}_{W} (- B F)$ and the bound $K$ of (18).

\begin{array}{ccccrrrr} B & ℓ & h & j_{ℓ + h} & W & F & R & K \\ 1 & 1 & 2 & 4 & 60 & 47 & 13 & 9 \\ 7 & 1 & 3 & 6 & 360 & 289 & 137 & 95 \\ 16 & 1 & 4 & 9 & 10800 & 8735 & 640 & 352 \end{array}

The first row reads as follows. The segment starts at $ℓ = 1$ and has length $2$ , so $W = b_{1} b_{2} = 6 \cdot 10 = 60$ ; the accumulated numerator is $F = 47$ ; and ${lpr}_{60} (- 47) = 13$ , since $- 47 + 60 = 13$ , which exceeds $K (1, 3) = ⌊ (16 + 40 + 27) / 9 ⌋ = 9$ . The third row lies outside the domain of (20), since $gcd (16, 30) = 2$ , and is displayed to illustrate the finite calculation at greater depth.

The larger scan covers $B \leq 5000$ coprime to $30$ and $100 \leq ℓ \leq 3000$ : all $3,869,934$ pairs satisfy the residue inequality, with first successful length at most $18$ in a search to length $24$ (saved run). Its histogram in Section 13.4 is a finite observation that the product-growth bound does not predict. The accompanying integer-only check reconstructs the blocks and tests all $2496$ pairs with $1 \leq B \leq 97$ , $gcd (B, 30) = 1$ and $1 \leq ℓ \leq 96$ . Every pair satisfies the residue inequality; the first successful lengths range from $1$ to $10$ , with a search limit of $24$ . This check uses the enumeration and finite recursions above, without floating-point logarithms. Neither computation proves the residue inequality for unbounded $B$ or arbitrarily late starts; Corollary 10.5 rules out covering the latter quantifier with any fixed maximum length.

Finite denominator bounds

A denominator bound from twelve blocks.

We obtain a denominator bound by enclosing $X_{1} = S - 1$ between two rational numbers. Here $h_{1} = 1$ , and summation through block $11$ , followed by the bound $Q (n_{12}) = Q (24) = 262 / 3$ with $P_{12} = 27993600000$ , gives

X_{1} = 2 \sum_{a = 1}^{11} \frac{m_{a}}{P_{a + 1}} + \frac{2 X_{12}}{P_{12}},

\frac{11327881097}{13996800000} < X_{1} \leq \frac{33983643553}{41990400000} .

The enclosure is strictly inside the interval $(11498 / 14207, 6409 / 7919)$ , as integer cross-multiplication verifies. The endpoint determinant is $14207 \cdot 6409 - 11498 \cdot 7919 = 1$ . For any rational $u / v$ strictly between them, with $v > 0$ , both integers $14207 u - 11498 v$ and $6409 v - 7919 u$ are positive. Consequently

v = 7919 (14207 u - 11498 v) + 14207 (6409 v - 7919 u) \geq 22126.

Thus, if $S$ is rational, its reduced denominator is at least $22126$ . This is the elementary denominator bound between two fractions with cross-determinant one, often expressed using Farey neighbours. It is also the best lower bound on a possible denominator obtainable from this enclosure alone: the mediant

\frac{11498 + 6409}{14207 + 7919} = \frac{17907}{22126}

is in lowest terms and lies strictly inside the certified enclosure. Thus the interval alone remains compatible with denominator $22126$ ; we have neither identified $X_{1}$ with this mediant nor determined a denominator for $S$ .

The certificate uses the actual block coefficients and proved tail bound. A narrower certified enclosure excludes further denominators; the continued-fraction exclusion below uses one built from $15,500$ blocks.

A denominator bound from $128$ consecutive blocks.

A finite lattice computation, outside the theorem inputs, excludes the following denominators. Using segments of length $128$ and $64$ starting indices, the first $10005$ , no rational value of the ${2, 3, 5}$ running-LCM series has reduced denominator $M B$ with $M$ a divisor supported on ${2, 3, 5}$ of $2^{10005} 3^{6312} 5^{4308}$ , $gcd (B, 30) = 1$ and $1 < B \leq B_{max}$ , where $B_{max}$ is the $106$ -digit integer

\begin{aligned} B_{max} = & 1134599670999687767349520845707093359257353022286558739363600235 \\ 016103207564063373270305324172145281971729, \end{aligned}

so that $\log_{2} B_{max} = 348.9846 \dots$ .

The exponent triple $(10005, 6312, 4308)$ is that of $H (2^{10005})$ . The normalisation therefore uses the full running LCM at its first start, whereas Lemma 9.2 uses the half running LCM $h_{10005}$ . This family of smooth denominators is the larger one.

The certificate and its exact checker are published. Its starts $a_{1} < \dots < a_{64}$ are the first $a \geq 10005$ with $W_{a, 128} = W_{10005, 128} = W = 2^{128} 3^{81} 5^{56}$ , so $\log_{2} W = 386.40993 \dots$ ; put $m_{0} = W / 2$ , $F_{j} = F_{a_{j}, 128}$ and $w_{j} = (a_{j} + 128)^{2} + 6 (a_{j} + 128) + 11$ . The certificate lists an integer matrix $C = (c_{i j})$ with $\sum_{j} c_{i j} F_{j} \equiv 0 (\mod m_{0})$ and $\sum_{j} | c_{i j} | w_{j} \leq 184,560,461,211$ for every row $i$ , an integer matrix $E$ with $C E = E C = m_{0} I$ , and $B_{max} = ⌊ (W - 1) / 184,560,461,211 ⌋$ . If $B \leq B_{max}$ and integers $f_{j} = W s_{j} - 2 B F_{j}$ satisfy $0 < f_{j} \leq B w_{j}$ , each $\sum_{j} c_{i j} f_{j}$ is a multiple of $W$ of absolute value less than $W$ , hence zero, and invertibility of $C$ forces every $f_{j}$ to vanish, a contradiction. The weights bound the tails: since $n_{b} \leq 2.07 b$ , Theorem 9.1 gives $2 X_{b} \leq 2 Q (n_{b}) \leq b^{2} + 6 b + 11$ for every $b \geq 0$ . The reduction from a rational value $N / (M B)$ to the $64$ residue inequalities, with $s_{j} = 2 B X_{a_{j}}$ and $f_{j} = 2 B X_{a_{j} + 128}$ , uses the iterated recurrence at full running LCM and is not machine-checked.

Continued-fraction exclusion.

A finite computation, outside the theorem inputs, shows that neither $X_{1}$ nor $S$ is a fraction with reduced denominator at most $2^{22482}$ , about $10^{6768}$ . Let $W$ and $F$ be the product of the bases and accumulated numerator of blocks $1$ to $15,500$ . The tail bound used for the twelve-block calculation, at $Q (n_{15501}) = Q (31956) = 340480534 / 3$ , gives $F / W < X_{1} \leq (F + Q) / W$ . Every block is built with integers, and blocks $1$ to $30$ are also checked against a direct enumeration of the smooth numbers. The two endpoints share their first $13,540$ partial quotients, and the numbers with a given prefix of partial quotients form an interval [24]. The first $13,109$ digits, $a_{0} = 0$ included, define a cylinder whose endpoints have cross-determinant $1$ and strictly contain the enclosure, so every fraction in the enclosure has denominator at least the sum of the two endpoint denominators, a $22,483$ -bit integer. The enclosure, the common digits, the regeneration program and the exact checker are published. A separate continued-fraction probe (saved run) certifies $13,109$ partial quotients after $a_{0}$ and records a largest denominator of $22,483$ bits, a largest partial quotient of $129,114$ , a mean partial quotient of $23.4133$ , observed Gauss–Kuzmin frequencies $0.4208$ , $0.1665$ , $0.0917$ , $0.0575$ , $0.0391$ against the predicted $0.4150$ , $0.1699$ , $0.0931$ , $0.0589$ , $0.0406$ , and a Lévy constant of $1.18869$ against $π^{2} / (12 \log 2) = 1.18657$ . The predicted values are the almost-everywhere frequencies of Gauss’s law and Lévy’s almost-everywhere constant [23]. These statistics describe a finite prefix and do not bear on whether $X_{1}$ is a Liouville number, algebraic, or rational with a larger denominator.

The two exclusions concern different finite families: one uses a lattice at a fixed starting index and fixed smooth part, the other a continued-fraction enclosure at $a = 1$ . Both have published witnesses and exact checkers. Each excludes only its stated denominator range, and neither settles an instance of the problem.

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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