A prime fractal and global quasi-self-similar structure in the distribution of prime-indexed primes

Abstract

Let p(n) be the nth prime and p(p(n)) be the nth prime-indexed prime (PIP). The process of taking prime-indexed subsequences of primes can be iterated, and the number of such iterations is the prime-index order. We report empirical evidence that the set composed of finite-differenced PIP sequences of prime-index order k >= 1 forms a quasi-self-similar fractal structure with scaling by prime-index order. Strong positive linear correlation (r >= 0.926) is observed for all pairwise combinations of these finite-differenced PIP sequences over the range of our sample, the first 1.3 billion primes. The structure exhibits translation invariance for shifts in the index set of the PIP sequences. Other free parameters of the structure include prime-index order and the order and spacing of the finite difference operator. The structure is graphed using 8-bit color fractal plots, scaled across prime-index orders k = 1..6 and spans the first 1.3 billion primes.