Conjecture 1.8 — Langevin

Definitions and mathematical statement

Let $k\ge1$ and let $n_1<\cdots<n_k$ be positive integers. They are multiplicatively dependent if there exist integers $a_1,\ldots,a_k$, not all zero, such that \[ \prod_{i=1}^k n_i^{a_i}=1. \] This means joint dependence of the tuple, not necessarily pairwise dependence or minimal dependence of all its entries.

Langevin's conjecture as extracted is \[ \left(n_1<\cdots<n_k\text{ positive and multiplicatively dependent}\right) \quad\Longrightarrow\quad \exists p\text{ prime},\qquad n_1\le p\le n_k. \] Both interval endpoints are included. The assertion asks for a prime in the numerical interval, not necessarily a prime equal to an entry or dividing an entry.

The source does not explicitly require $k\ge2$ or $n_1>1$. Under the usual definition, $1$ creates multiplicative dependence, and the singleton tuple containing $1$ requires an exception to the literal prime-in-interval assertion. The intended exclusion or tuple-length convention must be checked rather than silently added. No further spacing, composite-entry, or minimal-dependence condition is supplied.

Short English statement

Must the interval from the smallest to the largest integer in a multiplicatively dependent increasing tuple contain a prime? The singleton-$1$ boundary case needs source qualification.

The supplied summary attributes the conjecture to Langevin as recorded by Waldschmidt, links it qualitatively to Grimm-type and Sylvester--Schur questions, and reports no located proof or disproof of the exact statement. The OPEN-TRIAGE label is a supplied assessment; provenance says a current primary-literature audit remains required.

Sources

• Local UnsolvedMath registry, permanent ID 8700007, source code catalogue entry, "Conjecture 1.8 --- Langevin." Registry proposer metadata names Michel Waldschmidt; the title and summary attribute the conjecture itself to Langevin. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Michel Waldschmidt, Open Diophantine Problems (2004), Conjecture 1.8, https://arxiv.org/abs/math/0312440; source-TeX extraction, accessed 2026-07-29.
• Grimm-type and Sylvester--Schur problems are mentioned in the supplied summary without separate primary references.
• Source-specific provenance: the survey presents the item as open; current primary-literature status audit and redistribution terms NEEDS_REVIEW for the publicly accessible author/publisher copy, separately from the dataset licence.