Explicit Infinite Families of Squares, Cubes, and Fourth Powers with No Equal Adjacent Decimal Digits

Abstract

We study three fixed-exponent classes separately: squares, corresponding to OEIS A050741/A050749; cubes, corresponding to A050742/A050750; and fourth powers, corresponding to A050743/A050751. For squares, we recast the known construction of Rodrigo (2017), based on the roots \(R_{9t}\). For cubes, the repdigit-cube family \((3R_n)^3\) is already recorded in earlier sources and in OEIS A392833; we prove here that every member of this family has no equal adjacent decimal digits. For fourth powers, we give the family \((3R_{9t})^4\). Writing \(R_n=(10^n-1)/9\) for the decimal repunit, we derive exact decimal block formulas for all three cases by elementary base-\(10^m\) arithmetic with controlled borrowing. Consequently, A050741/A050749, A050742/A050750, and A050743/A050751 each contain an explicit infinite subfamily.