six exponentials theorem

six exponentials theorem

Complex numbers

x

1

,

x

2

,

…

,

x

n

are

ℚ

-

linearly independent

if the only

rational numbers

r

1

,

r

2

,

…

,

r

n

with

r

1

⁢

x

1

+

r

2

⁢

x

2

+

⋯

+

r

n

⁢

x

n

=

0

are

r

1

=

r

2

=

⋯

=

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. 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-S

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.

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

Generated on Fri Feb 9 15:35:45 2018 by

LaTeXML