On a Quadratic Collatz Type Map Governed by Prime Factorization. A Conjectural Universal Cycle, Two Reduction Theorems and an Intrinsic Computational Barrier.

We study a map on the integers defined by when is prime or even and when is odd composite, where denotes a designated prime factor . Two variants arise according to the choice of . In Part , is the largest prime factor of while in Part , it is the smallest. Informally, the conjecture of this paper asserts that every integer greater than one in absolute value eventually enters the single twelve element cycle and remains there forever. Formally, we verify this for every integer under Part with no exception and no alternate cycle observed and we conjecture it holds for every integer in the domain but we do not prove it. Toward this conjecture, we prove two theorems. The first is exact rather than asymptotic. For any odd composite , repeated application of the largest prime factor reduction reaches a prime in precisely steps where counts the prime factors of with multiplicity. Each application removes exactly one element from the prime factorization multiset so the count decreases by exactly one per step and terminates uniquely at a prime. The second theorem follows from the first. Under the extension of primality to negative integers through , which is the only convention under which the conjecture is well posed on , every negative integer reaches a positive value within at most steps. This reduces the negative integer case of the conjecture entirely to the positive integer case. We also identify an obstruction to further verification that is structural rather than a matter of computing resources. The reduction step of requires the complete factorization of the input which is a problem for which no general sub exponential algorithm is known. Repeated application of the squaring branch can therefore produce integers whose factorization lies beyond any presently known method regardless of computing time available. Our verification required factoring intermediate values of up to digits and succeeded in every instance though with no guarantee that a harder instance does not arise beyond the tested range. For Part , this obstruction is severe enough to foreclose even a conjecture. Every trajectory examined exceeded digits within fewer than iterations without any value repeating. We are unable to characterize the long-term behavior of Part by any method available to us and as a result, Part is entirely open.