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In number theory, a Liouville number is a real number with the property that, for every positive integer, there exists a pair of integers (,) with > such that < | | <. The inequality implies that Liouville numbers possess an excellent sequence of rational number approximations.
This might seem to be a much stronger result than Liouville's theorem, but it is actually an easy corollary. If the image of f {\displaystyle f} is not dense, then there is a complex number w {\displaystyle w} and a real number r > 0 {\displaystyle r>0} such that the open disk centered at w {\displaystyle w} with radius r {\displaystyle r} has ...
Siegel's theorem on integral points; Six exponentials theorem; Skolem–Mahler–Lech theorem; Sophie Germain's theorem; Størmer's theorem; Subspace theorem; Sum of two squares theorem; Szemerédi's theorem
In conformal mappings, see Liouville's theorem (conformal mappings) In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem; In linear differential equations, see Liouville's formula; In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental
Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but also mathematical physics and even astronomy. He is remembered particularly for Liouville's theorem in complex analysis.
The Liouville lambda function, denoted by λ(n) and named after Joseph Liouville, is an important arithmetic function. Its value is +1 if n is the product of an even number of prime numbers , and −1 if it is the product of an odd number of primes.
which does not satisfy Liouville's theorem, whichever degree n is chosen. This link between Diophantine approximations and transcendental number theory continues to the present day. Many of the proof techniques are shared between the two areas.
The irrationality exponent or Liouville–Roth irrationality measure is given by setting (,) =, [1] a definition adapting the one of Liouville numbers — the irrationality exponent () is defined for real numbers to be the supremum of the set of such that < | | < is satisfied by an infinite number of coprime integer pairs (,) with >.