Our lunchtime group got into another math related discussion. I apologize in advance if this isn't a rigorous question, as none of us are professional mathematicians. This is the question:
Is it possible to write every counting number (1, 2, 3, etc.) as a sum of no more than two primes, or as the sum of 1 and one prime?
Can anyone suggest links to articles or discussions about this?
I found the Goldbach Conjecture, which is about even counting numbers. Is there anything else?
Certainly not every odd integer $n>1$ is the sum of two primes. Already $27$ is a counterexample. However, every odd $n>1$ is indeed the sum of at most five primes. This is a very nice result, see here. If we consider the other condition whether $n=1+p$ is possible, the situation is not better. In general, $n-1$ need not be prime. Even if we combine the two conditions, there are many counterexamples.