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| {{distinguish|Gelfond's constant}}
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| The '''Gelfond–Schneider constant''' or '''Hilbert number'''<ref>{{citation|authorlink=Richard Courant|authorlink2=Herbert Robbins|first1=R.|last1=Courant|first2=H.|last2=Robbins|title=What Is Mathematics?: An Elementary Approach to Ideas and Methods|publisher=Oxford University Press|year=1996|page=107}}</ref> is
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| :<math>2^{\sqrt{2}}=2.6651441426902251886502972498731\ldots</math>
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| which was proved to be a [[transcendental number]] by [[Rodion Kuzmin]] in 1930.<ref name=Kuzmin>{{cite journal |author=R. O. Kuzmin |title=On a new class of transcendental numbers |journal=Izvestiya Akademii Nauk SSSR, Ser. matem. |volume=7 |year=1930 |pages=585–597 |url=http://mi.mathnet.ru/eng/izv5316}}</ref>
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| In 1934, [[Aleksandr Gelfond]] proved the more general ''[[Gelfond–Schneider theorem]]'',<ref>{{cite journal |author=Aleksandr Gelfond |title=Sur le septième Problème de Hilbert |journal=Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et na |volume=VII |issue=4 |pages=623–634 |year=1934 |url=http://mi.mathnet.ru/eng/izv4924}}</ref> which solved the part of [[Hilbert's seventh problem]] described below.
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| ==Properties==
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| The [[square root]] of the Gelfond–Schneider constant is the transcendental number
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| :<math>\sqrt{2^{\sqrt{2}}}=\sqrt{2}^{\sqrt{2}}=1.6325269\ldots</math>
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| 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 <math>\sqrt{2}^{\sqrt{2}}</math> is rational, which proves the theorem, or it is irrational (as it turns out to be), and then <math>\left(\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}}=2</math> is an irrational to an irrational power that is rational, which proves the theorem. The proof is not [[constructive proof|constructive]], as it does not say which of the two cases is true, but it is much simpler than [[Rodion Kuzmin|Kuzmin's]] proof.
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| ==Hilbert's seventh problem==
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| {{main|Hilbert's seventh problem}}
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| Part of the seventh of [[Hilbert's problems|Hilbert's twenty three problems]] posed in 1900 was to prove (or find a counterexample to the claim) that ''a<sup>b</sup>'' 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<sup>√2</sup>.
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| 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<sup>√2</sup>. 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.<ref>David Hilbert, ''Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920''.</ref> But the proof of this number's transcendence was published by Kuzmin in 1930,<ref name=Kuzmin/> well within [[David Hilbert|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.
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| ==See also==
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| * [[Gelfond's constant]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| * {{cite book | last=Ribenboim | first=Paulo | authorlink=Paulo Ribenboim | title=My Numbers, My Friends: Popular Lectures on Number Theory | series=Universitext | publisher=[[Springer-Verlag]] | year=2000 | isbn=0-387-98911-0 | zbl=0947.11001 }}
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| * {{cite book | editor=Felix E. Browder | editor-link=Felix Browder | title=Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume=XXVIII.1 | year=1976 | publisher=[[American Mathematical Society]] | isbn=0-8218-1428-1 | first=Robert | last=Tijdeman | authorlink=Robert Tijdeman | chapter=On the Gel'fond–Baker method and its applications | pages=241–268 | zbl=0341.10026 }}
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| {{DEFAULTSORT:Gelfond-Schneider Constant}}
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| [[Category:Transcendental numbers]]
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| [[Category:Mathematical constants]]
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