Stability of the Solar System ?
(Question often attributed to Newton in Opticks, 1717 or 1730.)
To further specify as requested by Timothy Chow, make it a few ($3\leqslant N\leqslant 8$) planets under pure Newtonian attraction, as in Suzuki, A history of the stability problem in celestial mechanics, from Newton to Laplace (1642–1787) (p. 24):
The other form of the stability question was that the planets themselves, by mutual gravitation alone, might disturb their orbits “until the system wants a reformation,” a point most famously raised by Newton in Optics. Because of the uncertainty of the validity of universal gravitation, this problem could not even be asked until after mid-century, but thereafter, progress was rapid and by 1760, a preliminary answer was obtained by Charles Euler, followed, over the next two decades, by the works of Lagrange and Laplace.
A “clear record of the problem being formulated as a conjecture or question” is in e.g. Laplace (1784):
si l’on n’a égard qu’aux lois de la gravitation universelle, les moyennes distances des corps célestes aux foyers de leurs forces principales sont immuables (…). Mais les excentricités et les inclinaisons sont-elles renfermées constamment dans d’étroites limites? C’est un point important du système du monde qui reste encore à éclaircir.
