A COMBINED CLASS OF PRIME NUMBERS ASSOCIATED WITH A PRIME TRIPLET

We introduce and study a combined family of nine exponential expressions attached to a triplet of distinct primes (p, q, r): the six multiplicative forms A^n*B +C and the three additive forms A^+ B*C obtained by permuting the parameters. A parity obstruction shows that a triplet of odd primes is never productive, so that the presence of 2 is necessary for any prime production. For triplets of the form (2, q, r), we establish four exact algebraic identities between the three base- 2 forms; these identities determine the relative order of the forms, govern their gaps, and lead naturally to Mersenne and Fermat primes as cases of minimal gap. We prove two twin-prime criteria of Mersenne and Fermat type, showing that for the triplet (2, 3, r) with r = 2n ± 1 prime and distinct from 3, two specific forms of the family differ by exactly 2, so that a twin prime pair arises as a structurally forced consequence of an algebraic identity. We then classify the gaps that the base-2 pair can realize: an even gap g can occur only if q− 1 divides g and the 2-adic valuations of q− 1 and of g coincide, and this condition becomes sufficient as soon as a prime of the form r = 2^n ± m exists, where m = g/(q− 1); this yields cousin and sexy prime configurations for explicitly determined values of q and excludes the gap eight altogether. We further establish a necessary-and-sufficient criterion of permanent obstruction modulo a single prime, for both multiplicative and additive forms, and prove that infinitely many pairs of distinct odd primes (q, r) make the form 2^n*q+ r composite for every n ≥ 1 by virtue of a multi-prime covering system of Sierpiński type. We further show that, for every triplet, the nine forms are simultaneously composite along an arithmetic progression of indices. We close with explicit numerical examples and with conjectures and open questions concerning infinitude, simultaneous productivity, and possible extensions to higher arities.