Solving the Fundamental Equation \(y^{{\tau}(r)} = y\) in \(\mathbb{Z}_r\)

Solving the Fundamental Equation yτ(r)=yin Zr Marco Ripà 2025-12-22 Abstract. In this short note, we aim to provide a compact formula for the smallest odd integer τ(r)>1such that the fixed-point equation yτ(r)=yhas the maximal set of solutions in the commutative ring of r-adic integers. Let r > 1be an integer and consider the radix-rnumeral system. Let p1, . . . , pω(r)be the distinct prime divisors of r. Let r:=Qω(r) j=1 pqj j, where qjis a positive integer, so that rad(r):=Qω(r) j=1 pj denotes the product of the distinct prime factors of r. In this short note, we aim to provide a compact formula for the smallest odd integer τ(r)>1 such that the fixed-point equation yτ(r)=yhas maximal set of solutions in Zr:= lim ←−n Z rnZ. The mentioned fixed-point equations originally arose in the course of deriving explicit formulas for the constant congruence speed of integer tetration bases in radix-rnumeral systems (with r squarefree but non-prime). In particular, the need to understand the stabilization of the solution set of yτ=yin Zremerged from the explicit analysis of the cases y3=yfor r= 6 and y5=yfor r= 10, discussed in [1]. Let S(r)denote the cardinality of the aforementioned (maximal) set of solutions of yτ(r)=yin Zr(here, it is worth noting that the trivial solutions 0,1r, and −1rare guaranteed by the fact that τ(r)is odd and by hypothesis). In general, S(r) =    rad(r)if 2∤r 3·rad(r) 2if 2|r (1) holds. Now, let ¯τ(r)>1be the smallest odd integer such that, for every odd τ > 1, all solutions of yτ=yin Zralready occur as solutions of y¯τ(r)=y. Finally, we may compactly state that ¯τ(r) =    3if rad(r) = 2 1 + lcm{pj−1:1≤j≤ω(r), pj= 2 }if rad(r)≥3 .(2) References [1] Ripà, M. & Di Pietro, G. (2025). A compact notation for peculiar properties characterizing integer tetration, Zenodo (preprint). Available at: https://zenodo.org/records/18012880. 1