numbrilisel

Numbrilisel is a term used in speculative mathematics to denote a class of integers that exhibit stability under a specific digit-driven transformation. The concept is not part of established number theory but is discussed in thought experiments and in fictional expositions on dynamical systems on integers.

Let s(n) be the sum of decimal digits of n, and let p(n) be the product of its decimal digits (with the convention p(n) = 0 if any digit is zero). Define T(n) = |s(n) − p(n)|. A positive integer n is called numbrilisel if the iterates T^k(n) eventually reach a fixed point, and the least k with T^k(n) = T^{k−1}(n) is bounded by the number of digits of n. In other words, there exists a function B(n) ≤ D(n) such that T^B(n)(n) is a fixed point.

This framework yields that every single-digit number maps to 0 in one step (since s(d) = p(d) = d), hence is numbrilisel. For multiple digits, the behavior depends on the digits and their arrangement; many numbers converge quickly, while a minority require more steps. Researchers studying numbrilisel typically examine the distribution of convergence steps and how it changes under digit permutations or concatenation.

Origin of the term traces to a neologism used in speculative discussions of digit-based dynamical systems. It appears mainly in fictional or pedagogical contexts to illustrate how simple arithmetic operations on digits can produce convergent processes. There is no consensus in mainstream mathematics on the formal status or utility of numbrilisel beyond these illustrations.

Numbrilisel is thus presented as a conceptual tool for exploring fixed points and convergence in digit transformations rather than as an established mathematical object.