Palfree Squares in a Two-Parameter Family of Near-Repdigits

For integers \(1 \le d \le 9\), \(0 \le e \le 9\), and \(n \ge 1\), let\[N_{d,e}(n)=d\frac{10^n-1}{9}+(e-d).\]For \(n \ge 2\), this is the decimal integer consisting of \(n-1\) copies of \(d\) followed by \(e\). We classify exactly when \(N_{d,e}(n)^2\) is palfree, meaning that its decimal representation contains no contiguous palindromic factor of length at least two. The proof uses an exact base-\(10^9\) decomposition in which the variable part consists of two repeated block bands. A local criterion reduces palfreeness to the exclusion of factors of the forms \(aa\) and \(aba\), and a repeated-block lemma shows that the infinite part stabilizes in every residue class modulo \(9\). The remaining finite certificate is checked by exact integer arithmetic. For each of the \(90\) parameter pairs \((d,e)\) there is a set \(\mathcal R_{d,e} \subseteq \{0,\ldots,8\}\) such that, for every \(n \ge 4\), the square is palfree if and only if \(n \bmod 9 \in \mathcal R_{d,e}\). Exactly \(37\) parameter pairs yield infinitely many palfree squares, \(53\) yield only finitely many, and exactly nine pairs yield a palfree square for every \(n \ge 1\). The natural density of successful indices for a fixed pair is \(|\mathcal R_{d,e}|/9\).