Transcendental Number - Sketch of A Proof That e Is Transcendental

Sketch of A Proof That e Is Transcendental

The first proof that the base of the natural logarithms, e, is transcendental dates from 1873. We will now follow the strategy of David Hilbert (1862–1943) who gave a simplification of the original proof of Charles Hermite. The idea is the following:

Assume, for purpose of finding a contradiction, that e is algebraic. Then there exists a finite set of integer coefficients satisfying the equation:

and such that and are both non-zero.

Depending on the value of n, we specify a sufficiently large positive integer k (to meet our needs later), and multiply both sides of the above equation by, where the notation will be used in this proof as shorthand for the integral:

We have arrived at the equation:

which can now be written in the form

where

The plan now is to show that for k sufficiently large, the above relations are impossible to satisfy because

is a non-zero integer and has absolute value smaller than one.

is an integer because each term is an integer times a sum of factorials, which results from the relation

which is valid for any positive integer j by the definition of the Gamma function.

It is non-zero because for every a satisfying, the integrand in is times a sum of terms whose lowest power of x is k+1 after substituting x for in the integral (assuming k ≥ n). Then this becomes a sum of integrals of the form with, and (again, from the definition of the Gamma function) it is therefore a product of . Thus, after division by, we get zero modulo (k+1) (i.e. a product of (k+1)). However, the integrand in has a term of the form and thus . By choosing which is prime and larger than n and, we get that is non-zero modulo (k+1) and is thus non-zero.

To show that

for sufficiently large k

we construct an auxiliary function, noting that it is the product of the functions and . Using upper bounds for and on the interval and employing the fact

for every real number G

is then sufficient to finish the proof.

A similar strategy, different from Lindemann's original approach, can be used to show that the number π is transcendental. Besides the gamma-function and some estimates as in the proof for e, facts about symmetric polynomials play a vital role in the proof.

For detailed information concerning the proofs of the transcendence of π and e see the references and external links.