View source for Gelfond–Schneider constant

{{distinguish|Gelfond's constant}}

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 [[2 (number)|two]] to the [[exponentiation|power]] of the [[square root of two]]:
:2{{sup|{{sqrt|2}}}} ≈ {{val|2.6651441426902251886502972498731}}...
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>
In 1934, [[Aleksandr Gelfond]] and [[Theodor Schneider]] independently 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.

==Properties==

The [[square root]] of the Gelfond–Schneider constant is the transcendental number 
:<math>\sqrt{2^{\sqrt{2}}}=\sqrt{2}^{\sqrt{2}} \approx </math> {{val|1.63252691943815284477}}....

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 <math>\sqrt{2}^\sqrt{2}</math> is a 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}}=\sqrt{2}^{\sqrt{2}\times\sqrt{2}}=\sqrt{2}^2=2</math>
is an irrational to an irrational power that is a rational which proves the theorem.<ref>{{citation
 | last = Jarden | first = D.
 | journal = [[Scripta Mathematica]]
 | page = 229
 | title = Curiosa: A simple proof that a power of an irrational number to an irrational exponent may be rational
 | volume = 19
 | year = 1953}}.</ref><ref>{{citation
 | last1 = Jones | first1 = J. P.
 | last2 = Toporowski | first2 = S.
 | doi = 10.2307/2319091
 | journal = [[American Mathematical Monthly]]
 | mr = 0314775
 | pages = 423–424
 | title = Irrational numbers
 | volume = 80
 | year = 1973| issue = 4
 | jstor = 2319091
 }},</ref>  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.

==Hilbert's seventh problem==

{{main|Hilbert's seventh problem}}

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''&nbsp;&ne;&nbsp;0,&nbsp;1 and irrational algebraic ''b''.  In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2<sup>{{sqrt|2}}</sup>.

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>{{sqrt|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 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 and by Schneider.

==See also==
* [[Gelfond's constant]]

==References==
{{reflist}}

==Further reading==
* {{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 }}
* {{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 }}

{{DEFAULTSORT:Gelfond-Schneider Constant}}
[[Category:Real transcendental numbers]]
[[Category:Mathematical constants]]