An Improved Explicit Upper Bound for Smallest Primes in Arithmetic Progressions for Prime Moduli

This paper addresses a classical problem in analytic number theory: establishing explicit upper bounds for the smallest prime in an arithmetic progression p ≡ a (mod q). We focus on the special case where q is prime and a is a primitive root modulo q.Under these conditions, we prove that for any ε > 0, there exists an effectively computable constant C(ε) such that p ≤ C(ε) · q^(5/2 + ε). This represents an improvement over the best-known general Linnik constant of 5, reducing the exponent to 5/2 + ε when a primitive root is present.The manuscript is self-contained and includes detailed error analysis, explicit constant tracking, and numerical verification for small q. A discussion of potential extensions and limitations is also provided.