Goldbach’s Conjecture states that every even integer greater than can be written as sum of primes. This problem is notoriously difficult, and still open.
However, one can prove the following version of Goldbach’s Conjecture for the ring :
Proposition. Let be a positive integer. Given any even integer there exist primes and such that .
The proof will, in fact, show that infinitely many such primes and exist. The required ingredients are Dirichlet’s Theorem on Arithmetic Progressions, and the following lemma due to Schinzel (1958):
Lemma. Given positive integers and , there exists an integer such that .
Proof of Lemma. The case is trivial, so assume and write its prime factorization. We have where are distinct primes, are some positive integers. For each , we will show that there exists an integer such that . If , we can let . If is an odd prime, cannot divide both and (otherwise, it would divide their difference which is ); so if let , and if let . Now, by Chinese Remainder Theorem, one can find an integer such that for each . It is evident that . (If it is not evident, assume . Then some prime divisor of would divide . But it doesn’t.)
Proof of Proposition. From the lemma, we have for some positive integer . By Dirichlet’s Theorem, there exist primes and of the form and . Thus, which proves , as desired. Indeed, Dirichlet’s Theorem guarantees the existence of infinitely many such primes and .