Infinitude of a Greedy Sequence of Perfect Powers with Distinct Neighboring Digits

We study the increasing greedy construction represented by OEIS A175031. Starting from perfect powers whose neighboring decimal digits are distinct, after a term is chosen the next term is the least larger admissible perfect power whose first digit differs from the last digit of the current term. We prove that this greedy process never terminates. The proof uses two unbounded square families. Writing\[R_{9t}=\frac{10^{9t}-1}{9},\qquad t\ge 1,\]we show directly that both\[R_{9t}^2\qquad\text{and}\qquad(2R_{9t})^2\]have distinct neighboring decimal digits. Their first digits are respectively \(1\) and \(4\). Thus, whatever the last digit of a current greedy term may be, one of the two families supplies a larger admissible witness. Consequently the next greedy term always exists, and OEIS A175031 is infinite.