The PKD Conjecture-Vô Conjecture 5 series 1(VC5S1)

Let $p, d$ be positive integers with $\gcd(p, d) = 1$, and define $N = \max(p, d)$. Consider the function:

We propose the PKD Conjecture, which asserts that for every such pair $(p, d)$ there exists an integer $k$ satisfying $0 < k < f(N)$ such that the linear form $q = pk + d$ is prime.

This conjecture provides an explicit logarithmic-square bound for the occurrence of primes in arithmetic progressions. If true, it would represent a substantial strengthening of existing bounds for the least prime in a given progression.