Mersenne Numbers, Recursive Generation of Natural Numbers, and Counting the Number of Prime Numbers

Collatz Conjecture Redefinition on Prime Numbers

“…A previous set of the employed concepts and definitions is necessary before developing the main idea of the present study. The readers can peruse our previous papers on the Collatz conjecture [1] [2] [3] and our studies about some characteristic features of prime numbers [4] [5]. These can provide the previous main definitions and some mathematical background.…”

“…The present paper follows the published effort of the author to understand some aspects of the Collatz conjecture [1] [2] [3] and several characteristics of prime numbers [4] [5]. Collatz conjecture has been studied from many points of view and still is an active scientific subject; see, for example, references [6]- [28].…”

On the Expression of Composite Natural Numbers as Tensorial Products of Prime Natural Powers Vectors

“…Combinatorics can produce an infinite set of sets of this kind; think about the freedom to choose among the known ones (see, for example, references [2] [7] [9]) 10 6 different heaps of 10 prime numbers natural power vectors of dimension 11. Then the number of composite natural numbers in this set will be 10 17 elements.…”