Universal $\pm 1$ Congruence Speed Invariant in Any Numeral System

Universal ±1Congruence Speed Invariant in Any Numeral System Marco Ripà 2025-12-08 A self-contained statement of the most elegant constant congruence speed identity characterizing tetration in numeral systems with radix r > 2. Consider the radix-rnumeral system with r > 2(unless specified otherwise) and let a > 1,b>1, c>0,k≥0be integers. Denote by νr(c)the maximum number of times rdivides c(i.e., νr(c) equals the largest integer msuch that rmdivides c, and νris a valuation when ris prime), and let rad(r) := Qp|rpdenote the product of the distinct prime factors of r. Let ba:= aa. . .a |{z} btimes be the height-btetration of a(e.g., 43 = 37625597484987). Let rbe given and assume that sbis the largest integer such that b+1a≡ba(mod rsb)and b+1a≡ ba(mod rsb+1). We define the radix-rcongruence speed of aat height bas V[r] b(a):=sb−sb−1. Consequently, for all integers b>1,c>1,k≥0, and t>νr(c)+1, the identity Vb[r] (k·rt+1 +rt−νr(c)+ 1)c=t=Vb[r] (k·rt+1 +rt−νr(c)−1)c(1) holds for all integers r > 2such that rad(r)∤cor r|c(i.e., the identity applies to every squarefree r > 2, and to all non-squarefree integers as well, except in the unique case where rad(r)divides c while rdoes not). Hence, we can compactly rewrite the universal identity (1) as Vb[r] (krt+1 +rt−νr(c)±1)c=t . (2) On the other hand, Vb[r] (k·rt+1 +rt−νr(c)±1)c≥t(3) holds for all integers r > 1,b > 1,c > 0,k≥0, and t > νr(c)+1. Since in (2) the “+1” case comes from the r-adic solution 1rof y3=ywhile the “−1” case corresponds to its symmetric solution −1r, and both hold in every ring of r-adic integers (r > 1), Vb[r]rt±1=V2[r]rt±1=t(4) is true in each radix-rnumeral system as long as rand tare integers greater than 1(we note that if t≥2, (4) is also true for the binary numeral system). If 3≤r=cand k= 0 are given, from (2), we conclude that Vb[r](rt−1+ 1)r=Vb[r](rt−1−1)r=t(5) is true for all integers b>1and t > 2. 1