...truth of the ABC conjecture would provide a new proof of [[Fermat's Last Theorem]].
...re", ed. Wüstholz, Gisbert; ''A panorama in number theory or The view from Baker's garden'', (2002), pp. 128-147, Cambridge University Press {{ZBL|1046.11035}
2 KB (362 words) - 19:28, 14 November 2023
...analogues and corollaries (cf. [[Thue–Siegel–Roth theorem|Thue–Siegel–Roth theorem]]; [[Diophantine approximations|Diophantine approximations]]). The non-effe
...N.I. Fel'dman, "An effective refinement of the exponent in Liouville's theorem" ''Math. USSR Izv.'' , '''5''' : 5 (1971) pp. 985–1002 ''Izv. Akad.
6 KB (940 words) - 18:12, 23 November 2014
...L| > \exp(-\epsilon B)$. The latter result was, however, only an existence theorem, and a bound for $B$, beyond which this inequality was satisfied, could not
...N.I. Fel'dman, "An effective refinement of the exponent in Liouville's theorem" ''Math. USSR Izv.'' , '''5''' : 5 (1971) pp. 985–1002 ''Izv. Akad.
5 KB (776 words) - 08:31, 23 November 2023
effective versions of Siegel's theorem, due to Baker, J. Coates and W.M. Schmidt); [[#References|[a2]]], Chapt. IV
to vary. The most celebrated example is Tijdeman's theorem on the Catalan equation [[#References|[a1]]], Chapt. 12, [[#References|[a3]
7 KB (1,087 words) - 19:41, 5 June 2020
...as proved by K. Weierstrass in 1885, is known as the Lindemann–Weierstrass theorem.
The method of proving Lindemann's theorem is known as the Hermite–Lindemann method. It is a development of Hermite'
3 KB (379 words) - 15:19, 19 August 2014
...istence proofs for a local Lie group with a given Lie algebra (Lie’s third theorem). Conversely, in any local Lie group, multiplication can be expressed in ca
6 KB (1,020 words) - 17:41, 4 May 2017
===Siegel's theorem on Dirichlet L-functions===
...lass number of a quadratic field of discriminant $-D$, it follows from the theorem that
5 KB (784 words) - 20:40, 18 October 2014
...s [[#References|[a11]]] gave a partial solution to Ulam's question. Hyers' theorem says that if a function $f : E _ { 1 } \rightarrow E _ { 2 }$ defined betwe
Following its appearance, Hyers' theorem was further extended in various directions (see [[#References|[a3]]], [[#Re
12 KB (1,800 words) - 19:48, 6 February 2024
...3]]], using the abstract [[Cauchy–Kovalevskaya theorem|Cauchy–Kovalevskaya theorem]]. The results for existence and for singularity formation use an extension
8 KB (1,185 words) - 17:00, 1 July 2020
can also be defined (Birkhoff's theorem) as a non-empty class of $ \Omega $-
10 KB (1,438 words) - 16:10, 1 April 2020
The above-mentioned theorem on the algebraic independence of the values of $ e ^{z} $
...bb{Q}$, is still (1986) unproved. An analogue of the Lindemann–Weierstrass theorem for values of the Weierstrass $ ( x _{1} \dots x _{n} ,\ e ^ {x _{1}} \d
6 KB (793 words) - 17:24, 17 December 2019
==Liouville's theorem on bounded entire analytic functions==
Liouville's theorem can be generalized in various directions. For example, if $ f (z) $
8 KB (1,240 words) - 04:55, 24 February 2022
By the Dirichlet unit theorem (cf. also [[Dirichlet theorem|Dirichlet theorem]]), the unit group $ U _ {F} $
6 KB (891 words) - 19:08, 26 March 2023
...eory of numbers]]). One of the first theorems of the theory was Khinchin's theorem [[#References|[1]]], [[#References|[2]]] which, in its modern form [[#Refer
...rm "almost-all" refers to Lebesgue measure in the respective space). The theorem describes the accuracy of the approximation of almost-all real numbers by r
8 KB (1,172 words) - 17:12, 8 March 2018
...legacyimages/d/d032/d032600/d03260069.png" />. It follows from Dirichlet's
theorem that for all real <img align="absmiddle" border="0" src="https://www.encycl
...2600/d03260080.png" /> (Kronecker's
theorem). An important feature of this
theorem on simultaneous inhomogeneous Diophantine approximations consists in the fa
54 KB (7,359 words) - 18:32, 31 March 2017
...f algebraic integers makes it possible to confirm the validity of Fermat's theorem for many classes of prime exponents $ n $.
...l be in contradiction with the [[Thue–Siegel–Roth theorem|Thue–Siegel–Roth theorem]], from which follows that the equation $ F( x, y)= C $,
18 KB (2,685 words) - 08:57, 18 August 2022
One of the first theorems in metric number theory is Borel's theorem (E. Borel, 1909): When written in an arbitrary fixed integer base $ g $,
...(cf. [[Normal number|Normal number]]). In an equivalent formulation, this theorem asserts that the fractional parts $ \{ \alpha g ^ {n} \} $,
19 KB (2,729 words) - 08:05, 14 January 2024
...valign="top">[7]</TD> <TD valign="top"> I.P. Mysovskikh, "On Chakalov's theorem" ''USSR Comp. Math. Math. Phys.'' , '''15''' : 6 (1976) pp. 221–227
12 KB (1,747 words) - 13:26, 14 January 2022
...y primes. Let $\pi(x)$ be the number of primes not exceeding $x$. Euclid's theorem can then be formulated as follows: $\pi(x)\to +\infty$ as $x\to\infty$. The
...absolute constant (cf. [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]). This problem was solved by methods of the theory of functions of a comp
27 KB (4,516 words) - 18:38, 18 October 2014
...{ \infty } s _ { k } z ^ { - k }$, which is called its symbol. Kronecker's theorem [[#References|[a9]]] says that $H$ has finite rank $n$ if and only if its s
As mentioned above (the Kronecker's theorem), if the Hankel operator $H$ has a rational symbol $r ( z ) = p ( z ) / q (
14 KB (2,126 words) - 16:45, 1 July 2020