Primes of the Form n~2 + n + p Have Density 1

摘要

In contrast with Sun's conjecture that any odd integer q > 3 can be written as n~2 + n + p with n a positive integer and p a prime, we consider the representation of primes in the form n~2 + n + p, where n is a positive integer and p is a prime. We prove that a subset of the primes having density 1 is representable in this form. We conjecture that every prime number > 5 is expressible in the form n2 + n + p where p is a twin prime, and we show that this conjecture implies the existence of infinitely many twin primes.