- Every even number greater than two can be written as the sum of two primes.

This conjecture has been researched by many number theorists and has been checked by computer for even numbers up to 4 × 10^{14}.
The majority of mathematicians believe that the conjecture is true, mostly based on statistical considerations focusing on the probabilistic distribution of prime numbers: the bigger the even number, the more "likely" it becomes that it can be written as a sum of two primes.
We know that every even number can be written as the sum of at most six primes, and in 1966, Chen showed that every sufficiently large even number can be written as the sum of a prime and a number with at most two prime factors.