Even-Digit Palindromes Divisible by (b + 1) and the Structure of the Quotient

Let b ≥3 be an integer base. Every even-digit palindrome P in base b is divisible by b + 1. We study the quotient P Q= b + 1, and show that for a large and natural class of palindromes the structure of Q is almost palindromic: there exists a canonical palindromic integer Qideal such that Q= Qideal + C j=1 ∆j , where the number of elementary corrections ∆j is at most the number C of monotonic- ity reversals in the first half of P and, for generic examples, equals it. Each correction ∆j is supported on a short interval that corresponds to a maximal “borrow chain” in the carry pattern of the division algorithm. We give complete proofs in arbitrary base b ≥3, discuss the binary case b = 2, and provide numerical experiments on several hundred thousand palindromes across multiple bases supporting the robustness of the phenomenon beyond the monotone–blocked class.