Adjacent-Digit Distinctness in Squares of Generalized Repunits

Abstract

For an integer base \(b \ge 3\), let\[R_n^{(b)}=1+b+\cdots+b^{n-1}=\frac{b^n-1}{b-1}\]be the length-\(n\) repunit in base \(b\). We prove that the standard base-\(b\) expansion of\[\left(R_n^{(b)}\right)^2\]has no two equal adjacent digits if and only if\[n=1\qquad\text{or}\qquadn\not\equiv1\pmod{b-1}.\] The proof follows the carries generated by the triangular convolution coefficients of the square and shows that the only possible equality of adjacent digits occurs at the central transition. Specializing to \(b=10\) gives an exact criterion for decimal repunit squares. In particular,\[\left(\frac{10^{9t}-1}{9}\right)^2,\qquad t\ge1,\]forms an infinite family of distinct elements of OEIS A090516, thereby proving the infinitude statement recorded for that sequence.