On the Finiteness and Asymptotic Behavior of Digital Root Digit Numbers.

For a fixed base b, we introduce a class of self-referential integers called Digital Root Digit (DRD) numbers. A positive integer n is a DRD number if n is equal to its digital root raised to its number of digits. We derive a structural equation characterizing DRD numbers and prove that only finitely many such numbers exist in every fixed base. Explicit bounds are obtained for the specific exponents associated with each digital root, yielding a complete finite search framework. Using asymptotic analysis, we show that the maximal persistence length grows on the order of b times natural logarithm of b as b goes to infinity. We further establish upper bounds for the total counting function and investigate its behavior across varying bases. Several open problems concerning monotonicity, asymptotic growth, and generalizations of the DRD framework are proposed