Similarities in Irrationality Proofs of ?, ln2, ?(2), and ?(3)

Introduction :

The irrationality of ? has dominated thousand ofyears of mathematical history, starting with thecircle-squaring problem of the ancient Greeks. In1761, Lambert proved the irrationality of ?(Lindemann completed with the transcendence proofin 1882). The interest in ?(3) started only a fewcenturies ago, but the number resisted until 1978,when R. Apery presented his « miraculous » proof.Even after Apery's discovery, scepticism stayedgeneral, until Beukers' simplified version grantedit. The character of the ?-numbers stillfascinates the mathematical community, includingstudents at ENS Lyon which think that thesenumbers are miraculous and have extraordinaryproperties.

Table of contents :

1) Four interesting numbers

The first two numbers which are ? and ln2, veryoften studied in high school. Their irrationnalityhas been prooved a long time ago, contrarily to?(3), defined by a serie in analysis. Thesenumbers are very different and a priori they arenot related.

2) Problem of rationnality

Though these numbers are easily defined, it isvery difficult to show their irrationnality. Thisquestion occupated a lot of mathematicians duringthe last two centuries.

a) Integral expression of these numbers

With tricky methods, we obtain an similar integralexpression of these numbers, which will be veryuseful. This way of reasoning is not intuitive butit is not really difficult when good elements areput together.

b) Four proofs in one

We will proove in the general case a theorem whichcould be used for numbers of a special form wherean unknowm function intervenes, and the fournumbers' irrationnality will be a consequence of it

3) Effective proofsWe give the functions associated to each number toconclude, using absurd reasoning, relevantmajorations and numbers theory. This way, wereally see the power of mathematics which mixesdifferent mathematical theories to obtain acrucial result.

Conclusion :

In this colloquium, we have explained a verygeneral method to show the irrationnality of  ?(3)with tricky ways. Unfortunately, it does not provide a proof for theirrationality of ? (4) (and by extension for ?(5)), which is an open problem.
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