The Goldbach conjecture says that every even integer greater than 2 is the sum of two primes. Nowadays we don’t count 1 as a prime, so 4 = 2 + 2 and 6 = 3 + 3. In other words, it seems the “two primes” are allowed to be the same.
Is there any even number greater than 6 that can’t be divided into two unequal primes?
never mind, I misunderstood
To the best of my knowledge, there are no such known examples. A separate branch of study has been to study how many unique ways an even number can be decomposed into a sum of two primes, and once you get to large enough numbers (3 or 4 digits), most have at least 10 different unique prime sum decompositions. There’s even an OEIS entry for it.
That’s not to say it is impossible, but it’s a much stronger statement than Goldbach’s conjecture itself to find an example that not only has one unique prime sum decomposition but for that decomposition to be twice a prime number. I suspect none exist but clearly, I do not have a proof.
At any rate, here is the Wiki on the topic:
Of course, even the Goldbach Conjecture itself isn’t known to certainly be true, so it should go without saying that this isn’t known certainly, either.
It’s a bit of an oddity but the Goldbach Conjecture doesn’t necessarily need to be true or false for the OP.
There may exist such an example either way.
I suspect this might become clearer if anyone ever proves the Riemann Hypothesis?
Hmm, what is Andrew Wiles doing these days…??
If there’s an even number that can’t be expressed as the sum of two primes, then it certainly can’t be expressed as the sum of two unequal primes. Or am I misinterpreting your comment?
Unless I’ve read it wrong, the OP is looking for an even number(s) larger than 6 that can only be expressed as the sum of two primes in one and only one way - the prime plus itself. Or else evidence such a number does not exist.
If I’m reading it right, there also seems to be an implicit assumption for the OP that the Goldbach Conjecture is true, but even that isn’t really necessary for such a number to exist, given what we know to date. Just because not every even number can be expressed as the sum of two primes does not mean there cannot still be an even number that can be so expressed in exactly one particular way.
If anybody has any proofs they’d like to share for any of that, please share!
Here’s a slight modification of the question:
Does there exist an even integer such the only way it can be written as the sum of two primes is n + n for n prime?
If such an integer existed, this would imply that for all even m < n, at least one of the integers n + m or n - m was composite. So a very closely related question would be the following:
Does every prime n form part of an arithmetic progression of primes with at least one prime before and after it?
If the answer to the second question is yes, then the answer to the first question is no, and vice versa. Arithmetic progressions of primes have been pretty extensively studied, but if this MathOverflow thread is to be believed, then it’s not known whether all primes are actually in any 3-term arithmetic progression, let alone in the middle of one.