Twin Primes In Quadratic Arithmetic Progressions

TL;DR

This paper rigorously proves the existence of infinitely many quadratic twin primes of the form n^2+1 and n^2+3, advancing the understanding of prime distributions in quadratic sequences.

Contribution

It provides a rigorous spectral method-based proof for infinitely many quadratic twin primes, building on previous heuristic and spectral analysis approaches.

Findings

Proof of infinitely many quadratic twin primes n^2+1 and n^2+3

Validation of spectral analysis approach for prime conjectures

Advancement in prime distribution in quadratic sequences

Abstract

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the spectral method, and a proof for the existence of infinitely many quadratic twin primes $n^{2}+1$ and $n^{2}+3$, $n \geq 1$, are proposed in this note.