|
Complex numbers $x_1, x_2, \ldots, x_n$ are $\Q$ -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 $\Q$ -linearly independent, and $y_1,y_2$ are also $\Q$ -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. Erdos [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.
- 1
- L. Alaoglu and P. Erdös, On highly composite and similar numbers. Trans. Amer. Math. Soc. 56 (1944), 448-469. Available online at www.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.
|
