In 1979 Beukers received his PhD at Leiden University under the direction of Robert Tijdeman with thesis The generalized Ramanujan–Nagell Equation, published in Acta Arithmetica, vol. 38, 1980/1981. From 1979 to 1980 he was a visiting scholar at the Institute for Advanced Study. He became a professor in Leiden and in the 2000s at Utrecht University.
Beukers works on questions of transcendence and irrationality in number theory, and on other topics. In connection with the famous proof by Roger Apéry (1978) on the irrationality of the values of the Riemann zeta function evaluated at the points 2 and 3, Beukers gave a much simpler alternate proof using Legendre polynomials. He also published on questions in mechanics about dynamical systems and their exact solvability.
- Differential Galois theory, in Michel Waldschmidt, Claude Itzykson, Jean-Marc Luck, Pierre Moussa (eds.): Number Theory and Physics, Les Houches 1989, Springer 1992
- Getaltheorie voor beginners (Number theory for beginners), Utrecht 2000
- A rational approach to Pi, Nieuw Archief voor Wiskunde, 2000, Heft 4
- A note on the irrationality of and . Bulletin of the London Mathematical Society, vol. 11, 1979, pp. 268–272