Primes of the form n^2+n+p have density 1

TL;DR

This paper investigates primes that can be expressed as the sum of a prime and twice a triangular number, proving that almost all primes have this form and linking the conjecture to the infinitude of twin primes.

Contribution

It establishes that a density 1 subset of primes can be represented in this form and connects the conjecture to the twin prime conjecture.

Findings

A density 1 subset of primes is expressible as prime plus twice a triangular number.

Conjecture that all odd primes are of this form implies infinitely many twin primes.

Provides a new perspective on prime representations and their relation to twin primes.

Abstract

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum of a twin prime and twice a triangular number. We show that this conjecture implies the existence of infinitely many twin primes.