Number of stable digits of any integer tetration

Abstract

In the present paper we provide a formula that allows to compute the number of stable digits of any integer tetration base a \in {\mathbb N}_0. The number of stable digits, at the given height of the power tower, indicates how many of the last digits of the (generic) tetration are frozen. Our formula is exact for every tetration base which is not coprime to 10, although a maximum gap equal to V(a)+1 digits (where V(a) denotes the constant congruence speed of a) can occur, in the worst-case scenario, between the upper and lower bound. In addition, for every a>1 which is not a multiple of 10, we show that V(a) corresponds to the 2-adic or 5-adic valuation of a-1 or a+1, or even to the 5-adic order of a^2+1, depending on the congruence class of a modulo 20.

On the constant congruence speed of tetration

“…As an example, let us take a = 807. Then, we have V (807, 1) = 0, V (807, 2) = 4, V (807, 3) = 4, V (807, 4) = 4, V (807, 5) = 4, and finally V (807, 6) = V (807, 7) = • • • = V (807) = 3 since ṽ5 (807 2 + 1) + 2 = 4 + 2 = 6 (by Definition 2.1 of Reference [5]) and 1 807 = 807; 2 807 ≡ 549620396283318273888501737943 (mod 10 30 ); 3 807 ≡ 601692651466822940525632857943 (mod 10 30 ); 4 807 ≡ 146336906474874632626032857943 (mod 10 30 ); 5 807 ≡ 355034907448973150626032857943 (mod 10 30 ); 6 807 ≡ 478635689812283150626032857943 (mod 10 30 ); 7 807 ≡ 027048888762283150626032857943 (mod 10 30 );…”

“…References [3][4][5] describe the value of V (a, b) for any a that is not multiple of 10. In particular, Equation (16) of Reference [5] gives the exact value of V (a) for all the mentioned tetration bases.…”

“…, while v 2 (m) and v 5 (m) indicates the 2-adic and the 5-adic valuation of m, respectively), the number of the rightmost digits of b a that are the same as those of b+1 a minus the number of the rightmost digits of b−1 a that remain the same by moving to b a is equal to the number of the new rightmost digits that freeze when we move from b+k a to b+k+1 a, for every k ∈ Z + (see Definitions 1.1 to 1.3 of Reference [5]). Now, for each non-negative integer a, let us indicate the number of new frozen digits at height b ∈ Z + as V (a, b), while V (a) denotes the (fixed) value of the constant congruence speed of a (i.e., the non-negative integer returned by V (a, b) for any b ≥ ṽ(a) + 2).…”