Gelfond–Schneider constant should not be confused with Gelfond's constant.
The Gelfond–Schneider constant or Hilbert number is two to the power of the square root of two:
2 ≈ ...which was proved to be a transcendental number by Rodion Kuzmin in 1930.[1] In 1934, Aleksandr Gelfond and Theodor Schneider independently proved the more general Gelfond–Schneider theorem,[2] which solved the part of Hilbert's seventh problem described below.
The square root of the Gelfond–Schneider constant is the transcendental number
\sqrt{2\sqrt{2
This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either
\sqrt{2}\sqrt{2}
\left(\sqrt{2}\sqrt{2
See main article: Hilbert's seventh problem.
Part of the seventh of Hilbert's twenty-three problems posed in 1900 was to prove, or find a counterexample to, the claim that ab is always transcendental for algebraic a ≠ 0, 1 and irrational algebraic b. In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2.
In 1919, he gave a lecture on number theory and spoke of three conjectures: the Riemann hypothesis, Fermat's Last Theorem, and the transcendence of 2. He mentioned to the audience that he didn't expect anyone in the hall to live long enough to see a proof of this result.[5] But the proof of this number's transcendence was published by Kuzmin in 1930,[1] well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent b is a real quadratic irrational, which was later extended to an arbitrary algebraic irrational b by Gelfond and by Schneider.
. Paulo Ribenboim . My Numbers, My Friends: Popular Lectures on Number Theory . Universitext . . 2000 . 0-387-98911-0 . 0947.11001 .
. Felix E. Browder . Felix Browder . Mathematical Developments Arising from Hilbert Problems . . XXVIII.1 . 1976 . . 0-8218-1428-1 . Robert Tijdeman . On the Gel'fond–Baker method and its applications . 241–268 . 0341.10026 .