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 primeP, one can say that $PM$ is P-even-like, and $PM+K$, withK = [1, 2, …, 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, …, 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 theP-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''.