On the Infinitude of Prime Tuples
Year: 2026
Type: preprint
Abstract: Dickson's conjecture and the Polignac conjecture both point to a central question: do prime tuples of a fixed admissible pattern occur infinitely often? This paper proposes an elementary argument based on the rigid structure of prime gaps, establishing the infinitude of arbitrary fixed admissible t-tuples. The core tool is the "Interval Pairing Principle": using the fact that the minimal prime gap is 2, we show that the partial products of the paired ratios do not collapse to zero. Combined with a "repeated sieve" mechanism, this yields a strict lower bound N(t) ≫ n/pm. Since the largest sieving prime pm= o(n), this lower bound diverges as n → ∞, completing the proof. The cases of twin primes and symmetric prime pairs (Goldbach-type estimates) are given as complete computational examples.
Author: Li, Qingming
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7166445801