Problem 458

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Statement. Let $[1,\ldots,n]$ denote the least common multiple of $\{1,\ldots,n\}$. Is it true that, for all $k\geq 1$,

$$[1,\ldots,p_{k+1}-1]< p_k[1,\ldots,p_k]?$$

Status. Falsifiable.

Source. erdosproblems.com/458, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #458, https://www.erdosproblems.com/458.

Formalization. Statement in formal-conjectures.

Current assessment

The site's label FALSIFIABLE, which the site explains as open but disprovable by a finite counterexample, marks the problem open and records the question's logical form: a counterexample is a single $k$ with $[1,\ldots,p_{k+1}-1]\ge p_k[1,\ldots,p_k]$, checked by finite arithmetic, while a proof must cover every $k$. The label is a body note, not a claim, and the problem has no claim page. The site's commentary records that Erdős and Graham expected the inequality to hold and saw two obstacles to a proof: ruling out several primes $q$ with $p_k<q^2<p_{k+1}$, which a gap bound of the strength of Legendre's conjecture would give, and the behavior of the small primes.

The site's discussion stated on 19 August 2025 that the comparison becomes a product over the prime powers strictly between two consecutive primes, each prime-power event contributing one factor of its base. A post of 27 April 2026 stated it again, counting a base once for each of its powers in the gap, and reported a check finding no counterexample with $p_{k+1}\le10^{12}$. A post of 12 June 2026 extends the check to $p_{k+1}\le10^{20}$. Below $4\times10^{18}$ it rests on the exhaustive gap search of Oliveira e Silva, Herzog and Pardi, Math. Comp. 83 (2014), and above that on the unrefereed distributed prime-gap search. As thread posts these have no claim page.

Search scope (2026-10-07): the site's page and its three-comment thread (2025-08-19, 2026-04-27 and 2026-06-12) and the formal-conjectures statement file.