On the infinitude of Prime $k$-tuples

TL;DR

This paper presents an inductive proof demonstrating that for each positive integer k, there exists at least one admissible set of k numbers that contains infinitely many prime k-tuples, extending Zhang's theorem.

Contribution

It introduces an inductive argument establishing the infinitude of prime k-tuples for at least one admissible set for each k, generalizing previous results.

Findings

Proves the existence of infinitely many prime k-tuples for some admissible sets.

Extends Zhang's theorem from prime doubles to arbitrary k-tuples.

Provides a new inductive framework for understanding prime k-tuple distributions.

Abstract

Starting with Zhang's theorem on the infinitude of prime doubles, we give an inductive argument that there exists an infinite number of prime $k$-tuples for at least one admissible set $\mathcal{H}_k=\{h_1,\ldots,h_k\}$ for each $k$.