Pi is transcendental

Pi is transcendental is a famous claim regarding the nature of the number $\pi$. It was first proven by Lindemann in 1882, buildung on methods developed by Hermite, who was able to prove that the number $e$ is transcendental roughly 10 years earlier in 1873. With the help of Weierstrass their findings could be generalized to the Lindemann-Weierstrass theorem in 1885.

Proof

The proof sketched out here is not the original proof by Lindemann but a later proof by David Hilbert which is more accessible.

Suppose $\pi$ is algebraic. Then ${\alpha }_1:=i\pi$ is also algebraic, so it is the root of an $n$-th degree polynomial with integer coefficients. Let ${\alpha }_2\text{, . . . , }{\alpha }_n$ be the other roots of this polynomial. Since $1+e^{i\pi }=0$ we have $(1+e^{{\alpha }_1})(1+e^{{\alpha }_2})\text{ . . . }(1+e^{{\alpha }_n})=1+e^{{\beta }_1}+e^{{\beta }_2}+\text{ . . . }+e^{{\beta }_N}=0$ with some algebraic ${\beta }_1\text{, . . . , }{\beta }_N$. Disregarding the ${\beta }_i$ that are equal to $0$, we end up with $a+e^{{\beta }_1}+e^{{\beta }_2}+\text{ . . . }+e^{{\beta }_M}=0$ for some positive integer $a$.

We now have to multiply both sides with an improper integral which I will just call $I_{\rho }$, with some positive integer $\rho$ as parameter. This will allow us to split the sum into two parts $P_1+P_2=0$, where $P_1$ and $P_2$ have the following properties:

• $P_1$ is an integer that can be divided by $\rho !$ and $\frac{P_1}{\rho !}\equiv a{b}^{\rho M+M}{b}_M^{\rho +1}\text{ mod }(\rho +1)$

where $b$, $b_M$ are the first and last coefficient of the polynomial with integer coefficients that has the ${\beta }_1\text{, . . . , }{\beta }_M$ as it's roots (so neither $a$, $b$ nor $b_M$ is zero).

• $|P_2|<\chi K^{\rho }$ for some positive constants $\chi \text{ and }K$.

So, if we choose $\rho$ big enough $\frac{P_2}{\rho !}$ tends towards zero while $\frac{P_1}{\rho !}$ always stays an integer. Furthermore the complicated congruence guarantees that $\frac{P_1}{\rho !}$ is not zero if $\rho$ is a multiple of $ab{b}_M$. Since $P_1+P_2=0$ it should also be the case that $\frac{P_1}{\rho !}+\frac{P_2}{\rho !}=0$. But that can't be when the first part get's arbitrarily small while the second part stays a non-zero integer.

Claim

Statement of the claim	Pi is transcendental
Level of certainty	Proven
Nature	Theoretical
Counterclaim	Pi is algebraic
Dependent on
Dependency of