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In order to have sum of $2$-primes to be a prime one of the primes must be the prime $2$. However the "distance" between adjacent primes increases as we search along the natural numbers.

For example The number of primes in the range: $10^5:10^5+100$ is $6$

The number of primes in the range: $10^7:10^7+100$ is $2$ $(10000019, 10000079)$

Is it possible we have finite numbers of this type?

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    $\begingroup$ en.wikipedia.org/wiki/Twin_prime $\endgroup$ – mathlove Apr 12 '16 at 6:50
  • $\begingroup$ It is conjectured to be infinite. I would say that due to the definition of increase, the statement the "distance" between adjacent primes increases as we search along the natural numbers is false. $\endgroup$ – barak manos Apr 12 '16 at 7:11
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    $\begingroup$ The distance between adjacent primes increases on average. You'll see the actual individual distances jump about quit a bit. For the first 25 primes the distances are 1,2,2,4,2,4,2,4,6,2,6,4,2,4,6,4,2,2,6,4,2,6,4,4,2,8. See how the numbers gradually get larger and larger numbers become more common but the small numbers never stop? $\endgroup$ – fleablood Apr 12 '16 at 8:20
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These are known as Twin Primes, and the question you are asking about is still unresolved, though I believe that most mathematicians suspect that their are indeed infinitely many such primes.

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