The $p$-adic Valuations of Möbius Duals of Lucas Sequences
Tyler Ross, Zhongyan Shen, Tianxin Cai
TL;DR
Extends Carmichael’s $p$-adic valuation results for Möbius duals of Lucas sequences to irregular cases by developing a framework for $M^U$ and $M^V$ and establishing a doubling relation $M^U_{2n}=M^V_n$ for odd $n$ (and $M^V_nM^U_n$ for even $n$). It derives comprehensive, case-by-case formulas for $v_p(M^U_n)$ in terms of the rank of apparition $z_U(p)$ and $p$-adic properties of $(P,Q,D)$, and demonstrates integrality properties: $M^U_n\in\mathbb{Z}$ for all $n$, while $M^V_{2n}$ is integral only for finitely many even indices. The results recover Carmichael’s regular-case valuations and place explicit bounds on when $M^V_{2n}$ can be integral (e.g., $n\le6$ if $D>0$, $n\le15$ if $D<0$). The work connects these valuations to Wall-Sun-Sun primes, showing an equivalence between their nonexistence and the squarefreeness of $M^F_n$ for $n\neq6$, and discusses known computational limits on such primes. Overall, it links Möbius-dual valuations, rank-of-apparition phenomena, and prime-entry behavior in Lucas sequences with potential implications for prime distribution in these recurrences.
Abstract
In this paper, we extend the $p$-adic valuations of the Möbius duals of Lucas sequences, originally obtained by Carmichael for regular Lucas sequences to irregular Lucas sequences. We conclude with a brief observation about the relationship of these valuations to the existence of Wall-Sun-Sun primes.