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Symmetry In Prime Numbers

Looking for symmetry in prime numbers does not necessarily mean "looking for symmetry IN prime numbers". Symmetry is found between them.

A hint of symmetry is displayed by the distance between them (the gap numbers)

Plotting consecutive gap numbers as a pair of coordinates reveals a symmetrical property:

A prime number is symmetrically surrounded by two other prime numbers only if their distance is a multiple of 6

 ∀(p, p+k, p + 2k), k=2×3×m, where m∈S={1/3,1,2,3,...,N}

Alternatively (preferred version):

if a < b < c are 3 consecutive primes, with a > 3,

a - 2b + c = 0

is true iff

b-a = c-b = 6n

with n ∈ ℕ.

In words: the sum of the two extreme terms minus twice the middle term equals zero iff their (equal) gap number is a multiple of 6. (excluding the first triple 3,5,7)

Here are some examples (consecutive):

47, 53, 59

151, 157, 163

167, 173, 179

251, 257, 263

257, 263, 269

367, 373, 379

557, 563, 569

This integer sequence is present in the On-line Encyclopedia of Integers Sequences: A128940