Gelfond–Schneider constant

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Not to be confused with Gelfond's constant.

The Gelfond–Schneider constant or Hilbert number[1] is

2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots,

which was proved to be a transcendental number by Rodion Kuzmin in 1930.[2] In 1934, Aleksandr Gelfond proved the more general Gelfond–Schneider theorem,[3] which solved the part of Hilbert's seventh problem described below.

Properties[edit]

The square root of the Gelfond–Schneider constant is the transcendental number

\sqrt{2^{\sqrt{2}}}=\sqrt{2}^{\sqrt{2}}=1.6325269\ldots.

This same constant can be used to prove that "an irrational to an irrational power may be rational", even without first proving its transcendence. The proof proceeds as follows: either \sqrt{2}^{\sqrt{2}} is rational, which proves the theorem, or it is irrational (as it turns out to be), and then \left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=\left(\sqrt{2}\right)^\left(\sqrt{2} \sqrt{2}\right)=\left(\sqrt{2}\right)^2=2 is an irrational to an irrational power that is rational, which proves the theorem.[4][5] The proof is not constructive, as it does not say which of the two cases is true, but it is much simpler than Kuzmin's proof.

Hilbert's seventh problem[edit]

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√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√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 final result.[6] But the proof of this number's transcendence was published by Kuzmin in 1930,[2] 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.

See also[edit]

References[edit]

  1. ^ Courant, R.; Robbins, H. (1996), What Is Mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press, p. 107 
  2. ^ a b R. O. Kuzmin (1930). "On a new class of transcendental numbers". Izvestiya Akademii Nauk SSSR, Ser. matem. 7: 585–597. 
  3. ^ Aleksandr Gelfond (1934). "Sur le septième Problème de Hilbert". Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na VII (4): 623–634. 
  4. ^ Jarden, D. (1953), Curiosa: A simple proof that a power of an irrational number to an irrational exponent may be rational, Scripta Mathematica 19: 229 .
  5. ^ Jones, J. P.; Toporowski, S. (1973), Irrational numbers, American Mathematical Monthly 80: 423–424, doi:10.2307/2319091, MR 0314775 ,
  6. ^ David Hilbert, Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920.

Further reading[edit]