Six exponentials theorem

Six exponentials theorem

In mathematics, specifically transcendental number theory, the six exponentials theorem is a result which, given the right conditions on the exponents, guarantees the transcendence of at least one of six exponentials.

tatement

If "x"1, "x"2, "x"3 are three complex numbers that are linearly independent over the rational numbers, and "y"1, "y"2 are two complex numbers that are also linearly independent over the rational numbers, then at least one of the following numbers is transcendental:

:e^{x_1y_1}, e^{x_1y_2}, e^{x_2y_1}, e^{x_2y_2}, e^{x_3y_1}, e^{x_3y_2}.

History

The theorem was first explicitly stated and proved in its complete form independently by Lang [S. Lang, "Introduction to transcendental numbers", Chapter 2 §1, Addison-Wesley Publishing Co., Reading, Mass., 1966.] and Ramachandra [K. Ramachandra, "Contributions to the theory of transcendental numbers. I, II.", Acta Arith. 14 (1967/68), pp.65-72.] in the 1960s. The theorem is weaker than the related but thus far unproved four exponentials conjecture.

Notes

External links

*MathWorld|urlname=SixExponentialsTheorem|title=Six exponentials theorem
*planetmath reference|id=4348|title=Six exponentials theorem


Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Four exponentials conjecture — In mathematics, specifically transcendence theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. tatementIf x 1, x 2… …   Wikipedia

  • 6 (number) — This article is about the mathematical number. For other uses, see 6 (disambiguation). 6 −1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Colossally abundant number — In mathematics, a colossally abundant number (sometimes abbreviated as CA) is a natural number that, in some rigorous sense, has a lot of divisors. Formally, a number n is colossally abundant if and only if there is an ε > 0 such… …   Wikipedia

  • Matrix (mathematics) — Specific elements of a matrix are often denoted by a variable with two subscripts. For instance, a2,1 represents the element at the second row and first column of a matrix A. In mathematics, a matrix (plural matrices, or less commonly matrixes)… …   Wikipedia

  • Logarithm — The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates (2, 1), (4, 2), and (8, 3) …   Wikipedia

  • Pattern theory — Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery …   Wikipedia

  • History of algebra — Elementary algebra is the branch of mathematics that deals with solving for the operands of arithmetic equations. Modern or abstract algebra has its origins as an abstraction of elementary algebra. Historians know that the earliest mathematical… …   Wikipedia

  • Rotation matrix — In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian… …   Wikipedia

  • Proofs of trigonometric identities — are used to show relations between trigonometric functions. This article will list trigonometric identities and prove them.Elementary trigonometric identitiesDefinitionsReferring to the diagram at the right, the six trigonometric functions of… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”