The Painlevé Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a positive integer $n$, consider positive masses $m_1,\ldots,m_n$ and positions $q_i(t)\in\mathbb R^3$ obeying Newtonian gravity, with the gravitational constant normalized to one: \[ \ddot q_i(t)=\sum_{j\ne i}m_j \frac{q_j(t)-q_i(t)}{\|q_j(t)-q_i(t)\|^3}. \] Initial positions are pairwise distinct and initial velocities are finite. A collision singularity at a finite terminal time is one at which the positions have finite limits with two of those limits coinciding. A noncollision singularity is a finite endpoint of a maximal classical solution that is not such a collision singularity; Newtonian examples have bodies escaping to unbounded distances in finite time. There are no collisions at earlier times of the classical solution.

Statement recorded in the supplied source.

For each integer $n\ge4$, do there exist positive masses and collision-free initial conditions for which the Newtonian $n$-body solution has a noncollision singularity at some finite time $T>0$? The spatial formulation includes planar motions. The historical existence question has been answered affirmatively; the cited four-body construction settles its final threshold case. [S2]

Short English statement

Can Newtonian point masses develop a finite-time singularity without a collision, for every number of bodies at least four? The cited work resolves the historical four-body case.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1142 (DYN-002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jinxin Xue, Noncollision Singularities in a Planar Four-body Problem, Acta Mathematica 224 (2020), 253--388, Section 1.1, Conjecture 2, and Section 1, Theorem 1; arXiv:1409.0048. https://arxiv.org/abs/1409.0048.