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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:

  1. All of them are finite
  2. All of them are infinite
  3. Some are finite and some are infinite

My intuition tells me that either one of the first two options is correct.

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My intuition says that there are only a finite number. What inspired this question? –  Display Name Aug 23 at 9:06
    
@DisplayName: I refrained from providing the motivation behind this question, in order to keep it as simple as possible. But you're right, it is a good point to mention it. Please see updated question. Thanks. –  barak manos Aug 23 at 9:27
    
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. –  TonyK Aug 23 at 9:43
    
@TonyK: Thank you very much. So the term of studies is known as Goldbach's Comet. Good to know that (and I would accept it as an answer if you posted it as such). It also means that there is a definite answer to at least one of my questions above - what is the largest known even number which is the sum of only one pair of prime numbers? –  barak manos Aug 23 at 9:51

2 Answers 2

up vote 5 down vote accepted

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.

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According to the answer made by @Travis, this has already been published at OEIS. –  barak manos Aug 23 at 12:27

It is conjectured that all of these are finite, but I don't know of any proofs in this direction. The largest (conjectured) even integers a_n that are expressible as a sum of two primes in n ways is exactly the content of https://oeis.org/A000954 (see https://oeis.org/A000954/b000954.txt for the sequence up to n = 5 000), which you could use to check your results of the easy programming exercise Tony recommended.

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