The Modulo-k Gilbreath Family: A Unified Generalization of Gilbreath Conjecture

This paper introduces and systematically studies a new family of Gilbreath-type iterated difference phenomena. For any odd integer \(k \geq 1\), we consider the sequence of all primes of the form \(kn + 2\). Repeatedly taking absolute differences of adjacent entries yields a difference triangle in which, from the second row onward, the leading entry eventually stabilizes to \(k\). This phenomenon constitutes an infinite family of structures parallel to the classical Gilbreath conjecture, termed the ``modulo-\(k\) Gilbreath family.'' The classical conjecture is recovered as the degenerate case when \(k=1\). Computational verification is provided for all odd \(k