A Relation between Prime Numbers : Extension of Gilbreath's Conjecture

We introduce a shifted difference operator on the sequence of prime numbers, extending the classical framework associated with Gilbreath’s conjecture. This operator depends on a fixed integer parameter m ≥ 1 and generates iterated difference sequences. We conjecture that the first element of these sequences eventually attains the value 1. Basic structural properties are established, and computational directions are discussed. But a thing to note is that this is a completely experimental paper. We have computed for finite primes and different shift.