On alternative definition of Lucas atoms and their p-adic valuations

Lucas atoms are irreducible factors of Lucas polynomials and they were introduced in Sagan and Tirrell (Adv Math 374:107387, 2020). The main aim of the authors was to investigate, from an innovatory point of view, when some combinatorial rational functions are actually polynomials. In this paper, we see that the Lucas atoms can be introduced in a more natural and powerful way than the original definition, providing straightforward proofs for their main properties. Moreover, we fully characterize the p-adic valuations of Lucas atoms for any prime p, answering to a problem left open in Sagan and Tirrell (2020), where the authors treated only some specific cases for p ∈ {2, 3}. Finally, we prove that the sequence of Lucas atoms is not holonomic, contrarily to the Lucas sequence that is a linear recurrent sequence of order two.