I am studying the special cases of Dirichlet theorem for infinitude of primes in arithmetic progressions, that is, $$ There \ are \ infinitely \ many \ primes \ that \ have \ the \ form \ of \ nk+1 $$ for a given natural number $n$
It is rather well-known that this statement has a non-analytic proof. ( such as http://mathoverflow.net/questions/32624/special-cases-of-dirichlets-theorem/32635#32635 )
Among such papers one assume the negative and make the enumeration of finite such primes like $p_1,p_2, ..., p_t$. But in order to make such enumeration, one should first prove that at least one prime of form $nk+1$ exists, in advance.
This is the point of my question. What do I have to prove the existence of prime of form $nk+1$ ?