six exponentials theorem

Complex numbers $x_{1},x_{2},\ldots,x_{n}$ are ${\mathbb{Q}}$ -linearly independent if the only rational numbers $r_{1},r_{2},\dots,r_{n}$ with

$r_{1}x_{1}+r_{2}x_{2}+\cdots+r_{n}x_{n}=0$

are $r_{1}=r_{2}=\cdots=r_{n}=0$ .

Six Exponentials Theorem: If $x_{1},x_{2},x_{3}$ are ${\mathbb{Q}}$ -linearly independent, and $y_{1},y_{2}$ are also ${\mathbb{Q}}$ -linearly independent, then at least one of the six numbers $\exp(x_{i}y_{j})$ is transcendental.

This is weaker than the Four Exponentials Conjecture.

Four Exponentials Conjecture: Given four complex numbers $x_{1},x_{2},y_{1},y_{2}$ , either $x_{1}/x_{2}$ or $y_{1}/y_{2}$ is rational, or one of the four numbers $\exp(x_{i}y_{j})$ is transcendental.

For the history of the six exponentials theorem, we quote briefly from [6, p. 15]:

The six exponentials theorem occurs for the first time in a paper by L. Alaoglu and P. Erdős [1], when these authors try to prove Ramanujan’s assertion that the quotient of two consecutive superior highly composite numbers is a prime, they need to know that if $x$ is a real number such that $p_{1}^{x}$ and $p_{2}^{x}$ are both rational numbers, with $p_{1}$ and $p_{2}$ distinct prime numbers, then $x$ is an integer. However, this statement (special case of the four exponentials conjecture) is yet unproven. They quote C. L. Siegel and claim that $x$ indeed is an integer if one assumes $p_{i}^{x}$ to be rational for three distinct primes $p_{i}$ . This is just a special case of the six exponentials theorem. They deduce that the quotient of two consecutive superior highly composite numbers is either a prime, or else a product of two primes.
The six exponentials theorem can be deduced from a very general result of Th. Schneider [4]. The four exponentials conjecture is equivalent to the first of the eight problems at the end of Schneider’s book [5]. An explicit statement of the six exponentials theorem, together with a proof, has been published independently and at about the same time by S. Lang [2, Chapter 2] and K. Ramachandra [3, Chapter 2]. They both formulated the four exponentials conjecture explicitly.

References

1 L. Alaoglu and P. Erdös, On highly composite and similar numbers. Trans. Amer. Math. Soc. 56 (1944), 448–469. Available online at http://links.jstor.org/sici?sici=0002-9947%28194411%2956%3A3%3C448%3AOHCASN%3E2.0.CO%3B2-Swww.jstor.org.
2 S. Lang, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass., 1966.
3 K. Ramachandra, Contributions to the theory of transcendental numbers. I, II. Acta Arith. 14 (1967/68), 65-72; ibid. 14 (1967/1968), 73–88.
4 Schneider, Theodor, Ein Satz über ganzwertige Funktionen als Prinzip für Transzendenzbeweise. (German) Math. Ann. 121, (1949). 131–140.
5 Schneider, Theodor Einführung in die transzendenten Zahlen. (German) Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. v+150 pp.
6 Waldschmidt, Michel, Diophantine approximation on linear algebraic groups. Transcendence properties of the exponential function in several variables. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 326. Springer-Verlag, Berlin, 2000. xxiv+633 pp. ISBN 3-540-66785-7.

Title	six exponentials theorem
Canonical name	SixExponentialsTheorem
Date of creation	2013-03-22 13:40:48
Last modified on	2013-03-22 13:40:48
Owner	Kevin OBryant (1315)
Last modified by	Kevin OBryant (1315)
Numerical id	5
Author	Kevin OBryant (1315)
Entry type	Theorem
Classification	msc 11J81
Synonym	6 exponentials
Related topic	FourExponentialsConjecture
Defines	linear independence