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

Intervals of constancy of the running LCM

We first describe which smooth integers contribute the same running value. The LCM is determined by the largest exponent of each prime in the prefix. For the unrestricted prefix this gives $lcm (1, \dots, N) = \prod_{t \leq N} t^{⌊ \log_{t} N ⌋}$ , with $t$ running over primes, and hence $\log lcm (1, \dots, N) = ψ (N)$ ; see Apostol [10] and Montgomery and Vaughan [12]. We apply the same exponent rule to a fixed prime support, keeping track of the intervals on which the running LCM is constant.

The smooth numbers up to $x$ are indexed by the exponent triples $(i, j, k)$ with $i \leq ⌊ \log_{p} x ⌋$ , $j \leq ⌊ \log_{q} x ⌋$ , $k \leq ⌊ \log_{r} x ⌋$ and $p^{i} q^{j} r^{k} \leq x$ . The coordinate bounds alone need not describe the prefix: each prime-power factor may be at most $x$ while their product exceeds $x$ . For instance, at $x = 6$ the factors $4$ , $3$ and $5$ satisfy their coordinate bounds, but the corresponding smooth number is $60$ .

Theorem 3.1 (the running least common multiple). Let $p, q, r$ be pairwise distinct primes and $x \geq 1$ . Then $L (x) = H (x)$ .

Proof. Every smooth $n \leq x$ has exponents bounded by the corresponding integer logarithms, so $n ∣ H (x)$ and hence $L (x) ∣ H (x)$ . Conversely the three pure powers $p^{⌊ \log_{p} x ⌋}$ , $q^{⌊ \log_{q} x ⌋}$ and $r^{⌊ \log_{r} x ⌋}$ are themselves smooth numbers not exceeding $x$ , so each divides $L (x)$ , and distinct primes have coprime powers, so their product divides $L (x)$ as well. ◻

At $(p, q, r) = (2, 3, 5)$ the first ten values are

\begin{array}{ccccccccccc} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ L (x) & 1 & 2 & 6 & 12 & 60 & 60 & 60 & 120 & 360 & 360 \end{array}

So $L (6) = 4 \cdot 3 \cdot 5 = 60$ , which exceeds $6$ : the running value at a smooth cutoff already contains powers of the other two primes that the cutoff itself does not. The kernel must therefore account for all three maximal powers, not just the factorisation of the cutoff. Also $H (x) \leq x^{3}$ , since each factor is at most $x$ . The exponent is the number of generating primes.

Say that $x$ and $y$ lie in the same logarithmic cell when $⌊ \log_{b} x ⌋ = ⌊ \log_{b} y ⌋$ for each of $b = p, q, r$ . By Theorem 3.1 the running value depends on $x$ only through the three integer logarithms, so it is constant on cells and moves only where one logarithm moves.

Proposition 3.2 (constancy and jump ratios). If $x, y \geq 1$ lie in the same logarithmic cell then $L (x) = L (y)$ , and the same holds for the kernel at two smooth points of one cell. If $⌊ \log_{p} y ⌋ = ⌊ \log_{p} x ⌋ + 1$ while the other two logarithms agree, then $L (y) = p L (x)$ , and similarly with $q$ or $r$ in place of $p$ .

Proof. Both parts are immediate from Theorem 3.1: the running LCM depends on $x$ only through the three integer logarithms, and advancing one of them multiplies exactly one factor by its base. ◻

Proposition 3.3 (jump count). Let $n \geq 0$ . The set of the first $n$ positive powers of $p$ , of $q$ and of $r$ has exactly $3 n$ elements, and adjoining the common origin $1$ gives exactly $3 n + 1$ .

Proof. For a fixed prime $b$ , the powers $b, b^{2}, \dots, b^{n}$ are distinct. A common value for two different primes would contradict unique factorisation. Finally $1$ is not a positive power of any prime. ◻

The later tail estimates use two elementary counting facts. Write $B (h_{p}, h_{q}, h_{r})$ for the exponent triples with $i \leq h_{p}$ , $j \leq h_{q}$ and $k \leq h_{r}$ , and $F (H)$ for the points of this box whose running LCM equals $H$ .

Proposition 3.4 (grouping equal running LCMs). For every box $B$ ,

\sum_{(i, j, k) \in B} K (i, j, k) = \sum_{H} # F (H) / H,

the outer sum ranging over the running LCMs attained on $B$ .

Proof. Partition $B$ into the fibres of the running LCM map. On $F (H)$ every summand is $1 / H$ , so the fibre contributes $# F (H) / H$ . ◻

Now fix an interval $[λ, η)$ with $0 \leq λ < η$ and write $S$ for the exponent triples of $B (h_{p}, h_{q}, h_{r})$ whose value $p^{i} q^{j} r^{k}$ lies in it. In the next lemma, the exponents $a, a^{'}$ are nonnegative integers, and $w \geq 0$ is the product of the fixed factors.

Lemma 3.5 (uniqueness in a short interval). Let $b \geq 1$ and $η \leq b λ$ . If $b^{a} w$ and $b^{a^{'}} w$ both lie in $[λ, η)$ then $a = a^{'}$ .

Proof. If $a < a^{'}$ then $η \leq b λ \leq b^{a + 1} w \leq b^{a^{'}} w < η$ , which is impossible; the case $a > a^{'}$ is symmetric. ◻

The short-interval condition says that multiplying by the omitted base moves a point beyond the interval. It holds for $[L, 2 L)$ when that base is at least two. A wider interval need not have this property: $1$ and $2$ both lie in $[1, 3)$ and have the same odd part. Thus the ratio bound, not just finiteness of the interval, permits the injective projection.

Proposition 3.6 (counting a block by two coordinates). If $η \leq r λ$ then $# S \leq (h_{p} + 1) (h_{q} + 1)$ , and if $η \leq p λ$ then $# S \leq (h_{q} + 1) (h_{r} + 1)$ . If moreover $η \leq r λ$ and $h_{p} \leq h_{q} \leq h_{r}$ with $h_{p} + h_{q} + h_{r} = j$ , then $9 # S \leq (j + 3)^{2}$ .

Proof. Suppose $η \leq r λ$ . If two triples of $S$ agree in their first two coordinates, Lemma 3.5 with $b = r$ and $w = p^{i} q^{j}$ forces their third coordinates to agree, so the projection forgetting the third coordinate is injective on $S$ and its image lies in a rectangle with $(h_{p} + 1) (h_{q} + 1)$ points. The other case is the same with the first coordinate projected away. Under the sorting hypothesis the two surviving coordinates are the two smallest, so it suffices that $a \leq b \leq c$ with $a + b + c = j$ gives $9 (a + 1) (b + 1) \leq (j + 3)^{2}$ . From $a \leq b \leq c$ we get $a + 2 b \leq j$ , so it is enough that $9 (a + 1) (b + 1) \leq (a + 2 b + 3)^{2}$ ; writing $b = a + d$ with $d \geq 0$ , the difference of the two sides is $d (3 a + 4 d + 3) \geq 0$ . ◻

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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