We solve a few open problems related to a peculiar property of the integer tetration ^{b}a, which is the constancy of its congruence speed for any sufficiently large b = b(a). Assuming radix-10 (the well known decimal numeral system), we provide an explicit formula for the congruence speed V(a) ∈ ℕ_0 of any a ∈ ℕ − {0} that is not a multiple of 10. In particular, for any given n ∈ ℕ, we prove to be true Ripà’s conjecture on the smallest a such that V(a) = n. Moreover, for any a ≠ 1 ∶ a ≢ 0 (mod 10), we show the existence of infinitely many prime numbers, p_j = p_j(V(a)), such that V(p_j) = V(a).

Integer tetration, the iterated exponentiation for ∈ ℕ − 0, 1 , is characterized by fascinating periodicity properties involving its rightmost figures, in any numeral system. Taking into account a radix-10 number system, in the book "La strana coda della serie n ^ n ^ ... ^ n" (2011), the author analyzed how many new stable digits are generated by every unitary increment of the hyperexponent , and he indicated this value as () or "congruence speed" of ≢ 0(mod 10). A few conjectures about () arose. If is sufficiently large, the congruence speed does not depend on , taking on a (strictly positive) unique value. We derive the formula that describes () for every ending in 5. Moreover, we claim that () = 1 for any (mod 25) ∈

In this paper, we present the clockwise-algorithm that solves the extension in 𝑘-dimensions of the infamous nine-dot problem, the well known two-dimensional thinking outside the box puzzle. We describe a general strategy that constructively produces minimum length covering trails, for any 𝑘 ∈ N−{0}, solving the NP-complete (3×3×⋯×3)-points problem inside a 3×3×⋯×3 hypercube. In particular, using our algorithm, we explicitly draw different covering trails of minimal length h(𝑘) = (3^𝑘 − 1)/2, for 𝑘 = 3, 4, 5. Furthermore, we conjecture that, for every 𝑘 ≥ 1, it is possible to solve the 3^𝑘-points problem with h(𝑘) lines starting from any of the 3^𝑘 nodes, except from the central one. Finally, we cover 3×3×3 points with a tree of size 12.

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.

In the present paper, we consider an optimization problem related to the extension in k-dimensions of the well known 3x3 points problem by Sam Loyd. In particular, thanks to a variation of the so called “clockwise-algorithm”, we show how it is possible to visit all the 3^k points of the k-dimensional grid given by the Cartesian product of (0, 1, 2) using covering trails formed by h(k)=(3^k-1)/2 links who belong to k (Euclidean) length classes. We can do this under the additional constraint of allowing only turning points which belong to the set B(k):={(0, 3) x (0, 3) x ... x (0, 3)}.