Algebraic Independence of Known Constants
Although both and e are known to be transcendental, It is not known whether the set of both of them is algebraically independent over . In fact, it is not even known if is irrational. Nesterenko proved in 1996 that:
- the numbers π, eπ, and Γ(1/4) are algebraically independent over Q.
- the numbers π, eπ√3, and Γ(1/3) are algebraically independent over Q.
- for all positive integers n, the numbers π, eπ√n are algebraically independent over Q.
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