six exponentials theorem
Complex numbers ${x}_{1},{x}_{2},\mathrm{\dots},{x}_{n}$ are $\mathbb{Q}$-linearly independent^{} if the only rational numbers ${r}_{1},{r}_{2},\mathrm{\dots},{r}_{n}$ with
$${r}_{1}{x}_{1}+{r}_{2}{x}_{2}+\mathrm{\cdots}+{r}_{n}{x}_{n}=0$$ |
are ${r}_{1}={r}_{2}=\mathrm{\cdots}={r}_{n}=0$.
Six Exponentials Theorem: If ${x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}{x}_{\mathrm{3}}$ are $\mathrm{Q}$-linearly independent, and ${y}_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{2}}$ are also $\mathrm{Q}$-linearly independent, then at least one of the six numbers $\mathrm{exp}\mathit{}\mathrm{(}{x}_{i}\mathit{}{y}_{j}\mathrm{)}$ is transcendental.
This is weaker than the Four Exponentials Conjecture^{}.
Four Exponentials Conjecture: Given four complex numbers ${x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}{y}_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{2}}$, either ${x}_{\mathrm{1}}\mathrm{/}{x}_{\mathrm{2}}$ or ${y}_{\mathrm{1}}\mathrm{/}{y}_{\mathrm{2}}$ is rational, or one of the four numbers $\mathrm{exp}\mathit{}\mathrm{(}{x}_{i}\mathit{}{y}_{j}\mathrm{)}$ 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 |