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 $13$.
Is every positive nonprime number at equal distance between two prime numbers?
4 Answers
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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1@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
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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0@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
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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1Copy + paste error, thanks for noticing my mistake. Re-edit time. – 2011-11-17
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.