Hilbert's seventh problem
- Hilbert's seventh problem
Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900. It concerns the irrationality and transcendence of certain numbers ("Irrationalität und Transzendenz bestimmter Zahlen"). Two specific questions are asked:
#In an isosceles triangle, if the ratio of the base angle to the angle at the vertex is algebraic but not rational, is then the ratio between base and side always transcendental?
#Is always transcendental, for algebraic and irrational algebraic ?
The second question was answered in the affirmative by Aleksandr Gelfond in 1934, and refined by Theodor Schneider in 1935. This result is known as Gelfond's theorem or the Gelfond–Schneider theorem. (The restriction to irrational "b" is important, since it is easy to see that is algebraic for algebraic "a" and rational "b".)
From the point of view of generalisations, this is the case
:
of the general linear form in logarithms which was attacked by Alan Baker.
ee also
*Hilbert number
External links
* [http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob7 English translation of Hilbert's original address]
Wikimedia Foundation.
2010.
Look at other dictionaries:
Hilbert's thirteenth problem — is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether or not a solution exists for all 7 th degree equations using functions of two arguments. It was first presented in the… … Wikipedia
Hilberts Liste von 23 mathematischen Problemen — Die hilbertschen Probleme sind eine Liste von 23, zum Zeitpunkt der Veröffentlichung, ungelösten Problemem der Mathematik. Sie wurden vom deutschen Mathematiker David Hilbert im Jahr 1900 beim Internationalen Mathematiker Kongress in Paris… … Deutsch Wikipedia
Liste von 23 mathematischen Problemen — Die hilbertschen Probleme sind eine Liste von 23, zum Zeitpunkt der Veröffentlichung, ungelösten Problemem der Mathematik. Sie wurden vom deutschen Mathematiker David Hilbert im Jahr 1900 beim Internationalen Mathematiker Kongress in Paris… … Deutsch Wikipedia
List of mathematics articles (H) — NOTOC H H cobordism H derivative H index H infinity methods in control theory H relation H space H theorem H tree Haag s theorem Haagerup property Haaland equation Haar measure Haar wavelet Haboush s theorem Hackenbush Hadamard code Hadamard… … Wikipedia
Gelfond–Schneider constant — The Gelfond–Schneider constant is :2^{sqrt{2=2.6651441...which was proved by Rodion Kuzmin to be a transcendental number. Aleksandr Gelfond in 1934 proved the more general Gelfond–Schneider theorem , which completely solved the part of Hilbert s… … Wikipedia
Gelfond's constant — In mathematics, Gelfond s constant, named after Aleksandr Gelfond, is :e^pi , that is, e to the power of π. Like both e and π, this constant is a transcendental number. This can be proven by Gelfond s theorem and noting the fact that: e^pi ; = ;… … Wikipedia
List of number theory topics — This is a list of number theory topics, by Wikipedia page. See also List of recreational number theory topics Topics in cryptography Contents 1 Factors 2 Fractions 3 Modular arithmetic … Wikipedia
Transcendental number — In mathematics, a transcendental number is a complex number that is not algebraic, that is, not a solution of a non zero polynomial equation with rational coefficients.The most prominent examples of transcendental numbers are π and e . Only a few … Wikipedia
Baker, Alan — ▪ British mathematician born August 19, 1939, London, England British mathematician who was awarded the Fields Medal in 1970 for his work in number theory. Baker attended University College, London (B.S., 1961), and Trinity College,… … Universalium
Satz von Gelfond-Schneider — Mithilfe des Satzes von Gelfond Schneider konnte zum ersten Mal eine umfangreiche Klasse von transzendenten Zahlen erzeugt werden. Er wurde zuerst 1934 von dem russischen Mathematiker Alexander Gelfond und unabhängig davon ein Jahr später von… … Deutsch Wikipedia
