<p> By the Dirichlet Theorem we know that the sequence $ latex ak+b$ has infinitely many primes. Assume that this sequence is $ latex (p_1, p_2, \ldots )$, where $ latex p_i= ak_i+b$. Can we find a bound $ latex k_{i+1} \leq f(a,b, k_i)$? Then knowing $ latex p_i$, we minimize $ latex f(a,b, k_i)$ with respect subject to $ latex p_i= ak_i+b$ to find its upper bound $ latex k^*$, yielding $ latex k_{i+1} \in [k_i+1, k^*]$. </p>