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For example $8$ is in the middle of the interval between $5$ and $11$, $9$ is at equal distance between $7$ and $11$; $10$ between $7$ and $13$.

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If so, then every even number is a sum of two primes. But this is a notorious open problem, known as the Goldbach conjecture.

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    @koijro, I think that the word *between* in the q$u$estion excl$u$des sa$y$ing that say 8 is at equal distance between 17 and 17. However one could argue that 3 is really not between 3 and 3 either (again it depends on language conventions).2011-11-24
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1 is a positive nonprime number not between any prime numbers at all. If you consider that cheating (I wouldn't know why), then see Gerry Myerson's anwer.

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    @Marcva$n$Leeuwe$n$: Origi$n$ally my comment was to a answer posted by AAA, and somehow that answer was later merged here (I assume sometime before you posted your comment to me). I guess them moderators were moderating (and my hat is off to them!)! Cheers!2012-07-28
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Check out a related theory: 'Green-Tao Theorem' which is a special case of Erdős conjecture and 'Primes in arithmetic progression' - in short, the primes contain arbitrarily long arithmetic progressions.

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    Copy + paste error, thanks for noticing my mistake. Re-edit time.2011-11-17
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Every prime number $>3$ also...

Every integer greater than 3 can be expressed as the average of two primes.

If a number is the average (or difference) of two primes, by doubling the number it has a partition of those two primes. So, for example, $(7+31)/2=19$ becomes $7+31=2∗19$. The Goldbach conjecture applies to even numbers only, but the average of two primes applies to every number - even, odd, prime - bigger than 3.

CSV of first 100,000: int, diff, p1, p2, type