Smale's 18 problems (1998)

Stephen Smale, Mathematical Intelligencer 20 (1998), 7–15; reprint in Arnold–Atiyah–Lax–Mazur, Mathematics: frontiers and perspectives (AMS, 1999). Written as a 21st-century update to Hilbert, at Arnold's IMU request.

Status cutoff 2026-08-17. Source: Wikipedia; Smale's essay.

#	Problem	Status	Note
1	Riemann hypothesis	Open	Also Hilbert 8a, Clay.
2	Poincaré conjecture	Resolved (yes)	Perelman 2003, Ricci flow.
3	P vs NP	Open	Clay.
4	Shub–Smale $\tau$ -conjecture (integer zeros of a univariate polynomial)	Open	Algebraic-complexity form of "P≠NP". Folder: `problems/smale-tau`.
5	Integer points on a plane curve in time $(2s)^c$	Open	Hilbert 10 is undecidable in many variables; this asks exponential time in two.
6	Finitely many relative equilibria in the $n$ -body problem?	Open	Almost all 5-body planar systems: Albouy–Kaloshin.
7	Fekete / log-energy points on $S^2$	Open	Related to Thomson. Finite- $N$ records are computational.
8	Dynamics in general-equilibrium price adjustment	Partial	Gjerstad 2013; Lindgren 2022. No consensus that this closes Smale's ask.
9	Strongly polynomial linear programming	Open	Feasibility of $Ax\ge b$ .
10	Pugh closing lemma, higher smoothness	Open	Hamiltonian surfaces: Asaoka–Irie.
11	Is 1-dimensional dynamics generally hyperbolic?	Split	(b) interval maps: Kozlovski–Shen–van Strien 2007. (a) complex / Mandelbrot: open.
12	Trivial centralizers dense in $\mathrm{Diff}^r(M)$ ?	Open	$C^1$ : Bonatti–Crovisier–Wilkinson. $r>1$ open.
13	Hilbert's 16th problem	Open	See `hilbert.md`.
14	Lorenz attractor is strange	Resolved (yes)	Tucker 2002, computer-assisted.
15	Navier–Stokes smoothness in $\mathbb R^3$	Open	Clay.
16	Jacobian conjecture	Split	$n=2$ still open. A 2026 claim (Alpöge, LLM-assisted) of a counterexample for $n=3$ (hence $n\ge 3$ ) is not a community consensus.
17	Approximate zeros of polynomial systems in average polynomial time	Resolved (yes)	Beltrán–Pardo 2008 (randomized); Lairez 2016 (deterministic).
18	Limits of intelligence	Too vague	Open-ended programme, not a yes/no.

Later addenda (Smale): mean-value problem; Gottschalk (is $S^3$ a minimal set?); Anosov vs Franks Lie-group model.

See also the Clay list in millennium.md (Smale 1, 3, 15).