Consider the set of all even numbers larger than $2$.
Goldbach's conjecture states that every element is the sum of a pair of prime numbers.
It has not been proved that all elements abide to that rule.
But it is trivial to show that there are infinitely many elements that do.
My question is as follows:
Are there infinitely many even numbers that are the sum of at most one pair of prime numbers?
If no, what is the largest known even number which is the sum of only one pair of prime numbers?
The general motivation behind this question is this:
Among all even numbers, some can be represented by only $1$ pair of prime numbers, some can be represented by $2$ different pairs of prime numbers, and so forth.
So we can split the infinite set of even numbers into disjoint subsets.
There are three options with regards to the cardinality of these subsets:
- All of them are finite
- All of them are infinite
- Some are finite and some are infinite
My intuition tells me that either one of the first two options is correct.
Best Answer
Heuristic reasoning (as well as numerical evidence) suggests that all of them are finite. Here is a discussion, with a couple of nice graphs. Here is a more focused article.
It looks like 12 is the largest even integer that can be expressed in only one way as a sum of two primes. Checking this up to some largish number (say, 1000000), and finding the corresponding maxima of your other subsets over that range, would make a nice programming exercise, which I might try when I have time.