Common nontrivial roots of Legendre polynomials

Determine whether two Legendre polynomials mathrm{Leg}_n(x) and mathrm{Leg}_m(x) can share a nontrivial root, thereby resolving the corresponding Stieltjes conjecture concerning common roots of Legendre polynomials.

Background

The paper discusses the arithmetic difficulty of the Legendre polynomial sequence Legn(x)\mathrm{Leg}_n(x). In particular, the authors note that even basic questions about irreducibility and common roots remain unresolved. The question of whether distinct Legendre polynomials share nontrivial roots is identified with the Stieltjes conjectures, named after Stieltjes's letter to Hermite.

References

A particular notorious case is that of Legendre polynomials $\mathrm{Leg}_n(x)$ for all $n$, where not only one cannot prove the irreducibility, but even the easier question of whether $Leg_n$ and $Leg_m$ might share nontrivial roots is open; these questions are known as Stieltjes conjecture(s), as they appear in Stieltjes's letter to Hermite .

— Generic Manin-Mumford  (2609.09354 - Bary-Soroker et al., 8 Sep 2026) in Remark immediately following Theorem \ref{thm:special-polynomials-intro}, Section 1