# Can every even integer greater than four be written as a sum of two twin primes?

Thinking of Goldbach conjecture I arrived at this

$\mathrm{Conjecture}$: Every even integer greater than four can be written as a sum of two twin primes.

What do you think?

I hope this is true. I tried to verify this up to some extent.

• @Bye_World $12=5+7$. Commented Mar 8, 2015 at 16:45
• I'm wondering if you mean that the two primes can be chosen from different twin primes sets. Commented Mar 8, 2015 at 16:52
• @user28111 :-) yes - I guess my comment was a suggestion to edit posed in rhetorical fashion. Commented Mar 8, 2015 at 16:55
• How can you express 42 as a sum of two twin primes? I didn't find any way but maybe... Commented Mar 8, 2015 at 17:55
• @Peterix 11+31 or 13+29 Commented Mar 8, 2015 at 18:23

In fact, it was already a conjecture; mathworld says, "It is conjectured that every even number is a sum of a pair of twin primes except a finite number of exceptions whose first few terms are $$2, 4, 94, 96, 98, 400, 402, 404, 514, 516, 518,\cdots$$" ... (OEIS A007534; Wells 1986, p. 132).

• Where does this conjecture come from, and why is the list of exceptions thought to be finite? (Mathworld only points to Wells' book, which I don't have access to.) Commented Mar 8, 2015 at 17:03
• I imagine the increasing count of partition into two primes as the even number increases in the general Goldbach conjecture means that the probability of having no such partition involving two twin primes falls quickly enough towards zero to make the finite claim plausible. Commented Mar 8, 2015 at 17:10
• @user28111: The OEIS page has a link to a one-page note by D. Zwillinger in 1979, which is earlier than Wells. Commented Mar 8, 2015 at 17:13
• @Elaqqad, is that J.J. Sylvester Stalone? ;-) Commented Mar 8, 2015 at 17:19
• @Elaqqad, I was just riffing on the funny typo in your first comment. Commented Mar 8, 2015 at 17:23

There are infinitely many even integers greater than four, so your conjecture would imply that there are infinitely many twin primes. Considering that the twin prime conjecture still has not been solved, I highly doubt that you will be able to prove your conjecture.

• the question asks if you can find a counter example for his conjecture because it's very strong then the Goldbach's conjecture or other conjecture so it will be easy to find a cotre-example, so maybe we can find a counter example but as you said we can not prove it and he doesn't asks us for a proof Commented Mar 8, 2015 at 16:48
• it turns out that you're wright and it's a very strong conjecture and there is no chance for proving such claims, but there is some finite exceptions. Commented Mar 8, 2015 at 17:14

I have found by the conjecture of goldbach that there are two unique prime numbers whose sum results in an even number . example- 64=53+11