Are There Exceptions to the Goldbach Conjecture for Even Numbers of the Form 2p?

[mathman]

4=2+2, 6=3+3. Are there any other cases where an even number of the form 2p, where p is a prime, cannot be represented as the sum of two different primes?

 

[CRGreathouse]

4=2+2, 6=3+3. Are there any other cases where an even number of the form 2p, where p is a prime, cannot be represented as the sum of two different primes?

There is almost surely no other case, but a proof is far beyond reach.

 

[dodo]

A quick&dirty computer program shows no cases up to a hundred thousand.

 

[CRGreathouse]

I can verify that there are no further exceptions up to http://www.research.att.com/~njas/sequences/A025018 (67) = 906030579562279642 ≈ 9 × 10¹⁷.

Edit: According to Tomás Oliveira e Silva, this bound is good up to 1.609 × 10¹⁸.

 

[blue_raver22]

The Goldbach conjecture, first proposed by Christian Goldbach in 1742, states that every even number greater than 2 can be expressed as the sum of two prime numbers. The variations mentioned, 4=2+2 and 6=3+3, are indeed valid cases where an even number can be represented as the sum of two different primes. However, there are other cases where an even number of the form 2p, where p is a prime, cannot be represented as the sum of two different primes.

For example, the even number 8 can only be expressed as 3+5, where 3 and 5 are both prime numbers. This is because the only other possible combination, 2+6, is not valid since 6 is not a prime number. Similarly, the even number 10 can only be expressed as 3+7 or 5+5. In fact, as the even numbers get larger, the number of possible combinations decreases significantly, making it difficult to find a pattern or generalization.

Despite numerous attempts and computer-assisted calculations, the Goldbach conjecture has yet to be proven or disproven. While there may be some patterns or variations, it is still an open problem in mathematics and continues to intrigue and challenge mathematicians. Therefore, it is important for scientists to continue researching and exploring the Goldbach conjecture and its variations in order to gain a better understanding of prime numbers and their properties.