On the fractal distribution of primes and prime-indexed primes by the binary image analysis

•The fractal nature of prime distribution is studied on binary images of primes.•The dimension of prime distribution is a non-integer value lower than 2.•The lacunarity of prime distribution of primes and PIPs have been computed.•The binary image for prime (and PIP) distribution is similar to a Cantor dust.

In this paper, the distribution of primes and prime-indexed primes (PIPs) is studied by mapping primes into a binary image which visualizes the distribution of primes. These images show that the distribution of primes (and PIPs) is similar to a Cantor dust, moreover the self-similarity with respect to the order of PIPs (already proven in Batchko (2014)) can be seen as an invariance of the binary images. The index of primes plays the same role of the scale for fractals, so that with respect to the index the distribution of prime-indexed primes is characterized by the self-similarity alike any other fractal. In particular, in order to single out the scale dependence, the PIPs fractal distribution will be evaluated by limiting to two parameters, fractal dimension (δδ) and lacunarity (λλ), that are usually used to measure the fractal nature. Because of the invariance of the corresponding binary plots, the fractal dimension and lacunarity of primes distribution are invariant with respect to the index of PIPs.