Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes or the form a + nd, where n is a non-negative integer. My question is concerning the composit terms of such arithmetic progressions, and particularly, concerning the prime factors thereof. Here are some examples and what I have found so far:
If the initial term is 1 and the common difference is 2, every prime other than 2 will appear as a factor of a composite term. I know that once a prime appears as a factor of a term, then the prime will continue to be a factor forevermore. My question is specifically concerning the initial appearance of the prime.
In any arithmetic progression of the form a + nd where a and d are any two positive coprime integers, and n is a non-negative integer, will every prime other than those which are factors of the common difference appear as a factor of a composite term?
If the initial term is 1 or 3 and the common difference is 2, I have found the answer to be yes. Likewise, if the initial term is 5 or 7, and the common difference is 6, the answer is yes. These I have checked by hand and have found proofs for.
Here is one specific example of what I am asking about:
If I set up the arithmetic progression with 2309 as the first term and use 2310 as the common difference, will I find a multiple of 13 within the first 13 terms? I know that the answer to this is yes. My question is, what is the proof for all such arithmetic progressions and the prime factors of the composit terms?