The problem
ζ(3) = Σ_{n≥1} 1/n³ is irrational. (More precisely: Apéry exhibited sequences converging fast enough to ζ(3) with denominators growing too slowly — the standard irrationality engine.)
Apéry's Theorem: ζ(3) Is Irrational
historicReferences
History & significance
After Euler's zeta(2)=\(pi^{2}\)/6 (1735) and the even values' pi-arithmetic, the odd values stonewalled everyone for nearly a quarter millennium.
The resolution (human proof)
Proven March 1978 by Roger Apéry, aged 64, presenting to a stunned Marseille audience; Cohen, Lenstra and van der Poorten reconstructed the argument in real time (van der Poorten's account is titled 'A proof that Euler missed'). The engine is a recurrence behind the Franel numbers producing approximations whose denominators grow too slowly for rationality; Beukers' later integral reinterpretation tied it to Legendre polynomials. Zeta(5) and beyond remain irrationality-open on our unsolved shelf.