Prove the irrationality of $0.235711131719...$ How can I prove that the number formed by concatenating the primes in order i.e. $0.235711131719...$ is irrational.
I know I have to demonstrate that it has no period, but I'll be so thankful if someone can explain very clear, including all cases.

Suppose this were to start repeating after some point with a period $N$. Beyond this point, for any $k$ there are at most $N$ different strings of length $k$ appearing in the repeating part. So it is enough to show that for large values of $k$  we have more than $N$ primes with $k$ digits. The prime number theorem tells us to expect the number of primes with $k$ digits to be on the order of $\frac{10^k}{k}$, so in particular for $k >> 0$ we can make this bigger than any $N$.
