Chapter 27 - A simple gasket derived from prime numbers

Abstract

Generally, well-known gaskets, such as the Sierpinski gasket, are fractal because increasingly larger voids are created as these structures evolve. As a counter-example, a gasket is constructed using prime numbers, which does not appear to have larger voids as it evolves. The prime number gasket is defined by the triangular array gij with (i) gij = fij, if fij = 0, and (ii) gij ≠ 1, if otherwise. The resulting structure appears superposed on a right-angled triangle when the gasket is terminated at some i = I. It is known that the number of primes less than a large integer n can be estimated as n/In(n). Prime numbers appear as ordered pairs; e.g., the twins (3,5) or (59,61), which are consecutive primes separated by a difference of 2, as quadruples (7, 11) or (13, 17), which are consecutive primes separated by a difference of 4, and so on. Hence, it may not be possible to analyze the presented gasket using number theory. The chapter gives a statistical analysis of this gasket with various illustarations.