This paper presents a proof of the Collatz conjecture for a specific subset of positive integers, those formed by multiplying a prime number ”p” greater than three with an odd integer ”u” derived using Fermat’s little theorem. Additionally, we introduce a novel screening criterion for identifying candidate twin primes, extending our previous work linking twin primes (p and p+2) with the equation 2(p−2) = pu+v, where unique solutions for u and v are required. This unified criterion offers a promising approach to twin prime identification within a wider range of integers, further advancing research in this mathematical domain