An Explicit Infinite Family of Fifth Powers with No Equal Adjacent Decimal Digits

Abstract

We study OEIS A050744/A050752, where A050744 records integers \(x\) for which the decimal expansion of \(x^5\) has no pair of equal adjacent digits and A050752 records the corresponding fifth powers. For every integer \(t\ge1\), we consider \[ x_t=\frac{2(10^{54t}-5\cdot10^{27t}+1)}{3}. \] We prove that \(x_t\) is an integer whose decimal expansion consists of \(27t-1\) copies of \(6\), followed by \(27t\) copies of \(3\), and a final digit \(4\), and that \(x_t^5\) has no equal adjacent decimal digits. The proof rewrites a fixed polynomial fifth power modulo \(3^5=243\) and converts the resulting identity into an exact base-\(10^{27}\) block concatenation with no hidden carries or borrows. Consequently, A050744 and A050752 each contain an explicit infinite subfamily.