A COMBINED CLASS OF PRIME NUMBERS ASSOCIATED WITH A PRIME TRIPLET

This paper introduces a unified, collective perspective to the study of parametrically generated prime numbers. Rather than analyzing isolated sequences, the author constructs and investigates a closed family of nine exponential expressions (six multiplicative and three additive forms) combinatorially attached to a single triplet of distinct primes (p,q,r). The paper first establishes a strict parity obstruction that eliminates any productivity for entirely odd triplets, proving that the presence of the prime 2 is structurally mandatory for prime production. Within the framework of triplets of the form (2,q,r), the study reveals a network of exact algebraic identities interlocking the base-2 forms. These structural relations dictate the relative order of the expressions, govern their gaps, and naturally bring forth Mersenne and Fermat primes as configurations of minimal gap. From these identities, the author derives two simultaneous primality criteria that structurally force the emergence of twin prime pairs. Finally, the paper characterizes the criteria for permanent modular obstruction modulo a single prime and demonstrates, via a multi-prime covering system of Sierpiński type, that infinitely many pairs (q,r) make certain forms composite for every integer n>=1. Explicit numerical applications illustrate the internal arithmetic resonance of this combined class, which opens new avenues at the interface of algebraic structures and analytic number theory.