Natural Numbers Generalized Parity, and the Collatz Algorithm

An extended definition of the even-odd property for natural numbers is used to highlightprime numbers in their classification. For a given natural number $M$ and a chosen prime$P$, one can say that $PM$ is $P$-even-like, and $PM+K$, with$K = [1, 2, \ldots, P-1]$, is $P$-odd-like of $K$-class. This approach enables thedescription of a generalized Collatz conjecture and an associated algorithm. A filtervector is used in a new $P$-Collatz algorithmic path to divide any number $M$ at any stepwhen it has prime factors also in the filter. If $M$ cannot be divided by any prime filtervector element, then the transformation $PM+K$ is performed. Therefore, a new formulationof the Collatz conjecture is: ``the $P$-Collatz algorithm converges to 1 for all naturalnumbers $M$, through a transformation of the form $PM+K$ (the original Collatz algorithmuses $P=3$ and $K=1$), provided that a conveniently chosen dimension $N$ of a filtervector $(2, 3, 5, \ldots, PN)$, built from a sequence of primes, is used to dividemultiples of its elements''. Given the $N$-dimensional filter vector, the prime $P$, andthe $K$-class, an initial natural number $M$ is called $P$-Collatz-compliant if the$P$-Collatz algorithm converges to 1 after a finite number of steps. Thus, the generalizedCollatz conjecture can be summarized as: ``Any natural number is $P$-Collatz compliant''.