Running least common multiples: distinct values and multiplicities

The historical and catalogue record

For a singleton $P=\{p\}$, we have $a_n=p^{n-1}$ and $[a_1,\dots ,a_n]=p^{n-1}$, so both sums equal $p/(p-1)$. Erdős’s 1988 account calls the infinite-prime case a simple exercise and presents irrationality for a finite number of primes greater than one as a probable extension [2]. Bloom’s catalogue records an open problem at the access date of 28 July 2026 [4]. These are dated source statements, not a fresh determination of the catalogue’s present status.

In the letter written on 1 January 1973 and published in 1974, Erdős calls the irrationality of ${\mathcal{R}}_P$ a conjecture that he cannot prove, and says that he can show the sum over the distinct values of $[a_1,\dots ,a_k]$ to be irrational [3]. That sum is ${\mathcal{D}}_P$. The letter introduces its primes as $p_1,\dots ,p_s$, later writing $p_r$, without restricting their number, so the singleton calculation forces the qualification $|P|\ge 2$. It prints no argument. The catalogue has cited the letter for Problem #269 since 28 December 2025, after the letter was quoted in the site’s discussion of missing problems [5], and its page records Erdős’s statement about the distinct values. The catalogue page says that Erdős proved the statement; the letter asserts it and prints no argument. Remark 5.1 proves it for every finite $P$ with $|P|\ge 2$. For $|P|=2$ it was already a consequence of the Hecke–Mahler transcendence theorem of Loxton and van der Poorten [16] and Bugeaud and Laurent [15], through the affine identity of Section 4, by the route Fan used for ${\mathcal{R}}_{\{p,q\}}$. We do not know what argument Erdős had in mind.

The literature search recorded here did not locate a proof of the assertion in print. Erdős and Graham state only the question for ${\mathcal{R}}_P$: the sum is irrational for infinite $P$, and they ask what happens for finite $P$ with more than one element [1]; the sum over distinct running LCMs does not appear there. Erdős’s survey of 1975 on the irrationality of series [8] does not mention the running least common multiple of the $P$-smooth integers, his survey of 1980 on combinatorial number theory [6] does not discuss irrationality, and his survey of 1988 states only the question for ${\mathcal{R}}_P$ [2]. The Hecke–Mahler transcendence theorems cited here, of Loxton and van der Poorten, Bugeaud and Laurent, and Luca–Ouaknine–Worrell [26], have a single slope; the identifications in Section 4 apply to ${\mathcal{D}}_P$ when $|P|=2$. The rationality criterion of Diananda and Oppenheim [7], which condenses a Cantor series into blocks of consecutive terms, has the form of the dyadic argument of Section 2; it restates rationality and does not decide ${\mathcal{D}}_P$. The discussion thread of Problem #269, as inspected for that search, supplied no argument for ${\mathcal{D}}_P$ with $|P|\ge 3$, and the Formal Conjectures statement [9] concerns ${\mathcal{R}}_P$ only. The Cantor-series criteria of Erdős–Straus [18] and Hančl–Tijdeman [19] do not apply to ${\mathcal{D}}_P$, as explained in Section 2. The search did not reach the sections on irrational series in Guy’s Unsolved Problems in Number Theory, or MathSciNet reviews. The unresolved target of this record is the repeated series ${\mathcal{R}}_P$; a transcendence statement for ${\mathcal{D}}_P$ with $|P|\ge 3$ is a separate question (Problem 12.7).

On 26 June 2026 Steve Fan posted the two-prime factorisation, the Hecke–Mahler reduction and the transcendence conclusion in the discussion thread of the problem’s page [17]; the comment itself notes that the argument does not seem to generalise immediately to $|P|\ge 3$, and a reply there observes that it applies to arbitrary coprime pairs. The first public version of this note is dated 22 July 2026, after Fan’s post. We retain the calculation as exposition and make no priority claim for it.

The statement of the problem has been formalised as a conjecture with an unfilled proof in the Formal Conjectures collection [9]. In the cited source, the rational, irrational and infinite-prime assertions end in sorry. Its Nat-indexed series includes the empty-prefix least-common-multiple term, so its value differs from the conventional one by a rational constant; transporting a theorem across that boundary needs an explicit series-identification lemma, which is not supplied here.