Is every positive nonprime number at equal distance between two prime numbers?

Solution 1:

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

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.

*Assuming the Goldbach conjecture

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