Complex numbers are -linearly independent if the only rational numbers with

Six Exponentials Theorem:If are -linearly independent, and are also -linearly independent, then at least one of the six numbers is transcendental.

This is weaker than the Four Exponentials Conjecture.

Four Exponentials Conjecture:Given four complex numbers , either or is rational, or one of the four numbers 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 is a real number such that and are both rational numbers, with and distinct prime numbers, then is an integer. However, this statement (special case of the four exponentials conjecture) is yet unproven. They quote C. L. Siegel and claim that indeed is an integer if one assumes to be rational for three distinct primes . 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.

Bibliography

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.