six exponentials theorem

Complex numbers x1,x2,,xn are -linearly independentMathworldPlanetmath if the only rational numbers r1,r2,,rn with


are r1=r2==rn=0.

Six Exponentials Theorem: If x1,x2,x3 are Q-linearly independent, and y1,y2 are also Q-linearly independent, then at least one of the six numbers exp(xiyj) is transcendental.

This is weaker than the Four Exponentials ConjectureMathworldPlanetmath.

Four Exponentials Conjecture: Given four complex numbers x1,x2,y1,y2, either x1/x2 or y1/y2 is rational, or one of the four numbers exp(xiyj) 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 p1x and p2x are both rational numbers, with p1 and p2 distinct prime numbersMathworldPlanetmath, 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 pix to be rational for three distinct primes pi. 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 productMathworldPlanetmathPlanetmathPlanetmath of two primes.
The six exponentials theorem can be deduced from a very general result of Th. Schneider [4]. The four exponentials conjecture is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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
  • 2 S. Lang, Introduction to transcendental numbersMathworldPlanetmath, 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 functionDlmfDlmfMathworld 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