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
The plan of attack now is to show that for k sufficiently large, the above relations are impossible to satisfy because
- is a non-zero integer and is not.
The fact that is a nonzero integer results from the relation
To show that
- for sufficiently large k
we first note that is the product of the functions and . Using upper bounds for and on the interval [0,n] 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.