# 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 $12=5+7$. – Vincenzo Oliva Mar 8 '15 at 16:45
• $12=5+7$ @Bye_World – user87543 Mar 8 '15 at 16:45
• Oops -- I was looking at a list of twin primes, then I realized that it only listed the first member of each pair. -- My bad. – user137731 Mar 8 '15 at 16:46
• I'm wondering if you mean that the two primes can be chosen from different twin primes sets. – Joffan Mar 8 '15 at 16:52
• @user28111 :-) yes - I guess my comment was a suggestion to edit posed in rhetorical fashion. – Joffan Mar 8 '15 at 16:55

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.) – Noah Schweber Mar 8 '15 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. – Joffan Mar 8 '15 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. – Nate Eldredge Mar 8 '15 at 17:13
• @Elaqqad, is that J.J. Sylvester Stalone? ;-) – Barry Cipra Mar 8 '15 at 17:19
• @Elaqqad, I was just riffing on the funny typo in your first comment. – Barry Cipra Mar 8 '15 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 – Elaqqad Mar 8 '15 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. – Elaqqad Mar 8 '15 at 17:14

EDIT: I misunderstood the question, as noted in the comments below.

Thinking out loud: one natural approach to disproving the conjecture would be to show that it would violate known upper bounds on the frequency of twin primes. One that leaps out to me is Brun's theorem http://en.wikipedia.org/wiki/Twin_prime#Brun.27s_theorem, that the sums of the reciprocals of the twin primes is convergent - keep in mind that the sum of the reciprocals of the primes is divergent.

EDIT: however, as Elaqqad's answer above indicates, it can't be this easy.

• $10=5+5$ and $26 =13+13$. OP never said that the two twin primes had to be distinct. I think you're misinterpretting the question. – user137731 Mar 8 '15 at 16:51
• the question does not require the two integers to be the successive primes, for example $14=7+7$ or $26=13+13$ or $20=13+7$ are all acceptable writings because the primes $7,13$ are twin primes – Elaqqad Mar 8 '15 at 16:51
• 28=11+17 - Grothendieck would be proud, though. :P – Noah Schweber Mar 8 '15 at 16:53