A Family of Hailstone Maps on the Primes

Author(s)	Mr. Najeem Ziauddin
Country	India
Abstract	We study prime-to-prime iterated maps in which the branch is selected by the residue of the current prime modulo six, with the output obtained by rounding a real or integral target to a nearby prime. Two motivating maps arise from OEIS entries A365048 and A367479. We embed the affine map underlying A365048 in a two-parameter family and organize it with square-root and hybrid variants in a common growth–decay framework. An exact criterion is proved for consecutive parameter values to agree at a given affine-branch state, together with a congruence condition under which this agreement holds unconditionally. We also obtain a necessary imbalance between affine and decay steps in any sufficiently large cycle. Finite computations extend and correct the reported parameter searches, compare stopping-time growth on a common range of starting primes, and exhibit explicit cycles for nearby modified maps. Universal termination is left as conjecture.
Keywords	Integer Sequences, Primes, Discrete Dynamics, Collatz-Type Maps, Prime Gaps.
Field	Mathematics
Published In	Volume 8, Issue 4, July-August 2026
Published On	2026-08-14
DOI	https://doi.org/10.36948/ijfmr.2026.v08i04.85803

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