Introduction to Diophantine Methods Irrationality and Trancendence

1 Irrationality 3 1.1 Simple proofs of irrationality . . . . . . . . . . . . . . . . . . . . 3 1.2 Variation on a proof by Fourier (1815) . . . . . . . . . . . . . . . 10 1.2.1 Irrationality of e . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 The number e is not quadratic . . . . . . . . . . . . . . . 11 1.2.3 Irrationality of e √ 2 (Following a suggestion of D.M. Masser) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.4 The number e is not quadratic . . . . . . . . . . . . . . . 13 1.2.5 The number e √ 3 is irrational . . . . . . . . . . . . . . . . 14 1.2.6 Is-it possible to go further? . . . . . . . . . . . . . . . . . 15 1.2.7 A geometrical proof of the irrationality of e . . . . . . . . 15 1.3 Irrationality Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Statement of the first criterion . . . . . . . . . . . . . . . 16 1.3.2 Proof of Dirichlet’s Theorem (i)⇒(iii) in the criterion 1.6 17 1.3.3 Irrationality of at least one number . . . . . . . . . . . . . 18 1.3.4 Hurwitz Theorem . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.5 Irrationality of series studied by Liouville and Fredholm . 26 1.3.6 A further irrationality criterion . . . . . . . . . . . . . . . 28 1.4 Irrationality of e and π . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1 Irrationality of e for r ∈ Q . . . . . . . . . . . . . . . . . 29 1.4.2 Following Nesterenko . . . . . . . . . . . . . . . . . . . . . 30 1.4.3 Irrationality of π . . . . . . . . . . . . . . . . . . . . . . . 33 1.5 Padé approximation to the exponential function . . . . . . . . . . 34 1.5.1 Siegel’s point of view . . . . . . . . . . . . . . . . . . . . . 34 1.5.2 Hermite’s identity . . . . . . . . . . . . . . . . . . . . . . 40