The Least Prime in Arithmetic an Progression

In his 1979 paper Samuel Wagstaff studied the problem of bounding the first prime in an arithmetic progression. In this paper we update a number of his computations using advances in hardware. Based on this we refine his conjecture on Primes in Arithmetic Progression and provide further numerical evidence in support of it. For instance we conjecture that for $n>3$ we have a bound of $3\phi (n)\log (n)\log (\phi (n))$ and verified this bound to ${10}^8$.