Is there a way to prove that there are infinite primes ending in 777?

Question

I am aware that it can be solved using Dirichlet's theorem, but are there are any other methods to prove it?

Answer 1

Here is some strong evidence that the answer is no:

In 1988, M. Ram Murty proposed a rigorous definition of a "Euclidean proof" that a particular arithmetic progression contains infinitely many primes (trying to characterize what kind of proof would look like Euclid's proof that there are infinitely many primes). Murty showed that such a proof exists for the residue class $a\pmod m$ if and only if $a^2\equiv1\pmod m$. (Here is a paper that gives an account of this topic and proof.)

The positive numbers ending in $777$ are precisely the numbers that are congruent to $777\pmod{1000}$. Since $777^2\not\equiv1\pmod{1000}$, there is no Euclidean proof of the infinitude of such primes.

Amusingly, Murty's result says that there is a Euclidean proof of the infinitude of primes ending in $999$. (!)