Dulac in 1923 claimed to have proven
Any real planar differential equation $$\frac{dx}{dt} = Q(x,y) \qquad \frac{dy}{dt} = P(x,y) $$ where $P,Q$ are polynomials with real coefficients, has a finite number of limit cycles.
His proof turned out to have a large hole. It took until the 1980s and the (independent) work of Écalle on resummation and the Borel-Laplace tranform, and the work of Ilyashenko on analytic continuation in the complex plane of the Poincaré first real return map associated to polycycles. Both of these strands of work are phenomenal achievements, and bore fruit for quite some time (maybe still does, but I stopped following this area some years back).
