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Roger Apery's seminal method for proving irrationality is "turned on itshead" and taught to computers, enabling a one second redux of the originalproof of zeta(3), and many new irrationality proofs of many new constants,alas, none of them is both famous and not-yet-proved-irrational.

Searching for Apery-Style Miracles [Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]