I asked my last question on the topic of representing numbers by decimals which was intimately linked to another matter: namely, the relationship between the size of a decimal expansion of a fraction and it's numerator/denominator. This in itself led to some other inquiries and then the following question popped up:
Is it true that there exist an infinite number of primes whose decimal string has no consecutive repeated digits?
This seems to be an even more difficult question and I am curious if one can hope of giving this an answer.
We can give a heuristic answer. Consider all the $n$ digit numbers. There are $9\cdot 10^{n-1}$ of them, of which $9^n$ have no repeated digits. The density of primes around $N$ is $\frac 1{\log N}$. We will take the maximum $N$ among our numbers, giving the minimum chance for a prime, which is $10^{n}$. We therefore expect more than $\frac {9^n}{\log \left(10^{n}\right)}$ primes in the decade with no repeated digits. This is huge compared with $1$, so summing over $n$ will give an infinite number of expected primes with no repeated digits.