Palindromic Differences

Let x denote a positive integer, written in the ordinary denary form, and define its palindromic inverse x T to be the integer obtained from x by writing its digits in reverse order. We ignore leading zeros so that both 1234 and 12340 have palindromic inverse 4321. A number is called a palindrome if x = x f. Similar definitions apply to bases other than 10. A notorious problem concerns palindromic sums [3]. From any starting point Xi we form a sequence inductively by x^ + i = Xy. + x£ , and the question is whether one always arrives at a palindrome. A negative answer is conjectured, and specifically that for Xi = 196 a palindrome is never reached. Although this problem is unsolved, the conjecture is known to be correct for base 2 [2]. The problem, however, is somewhat artificial since the property of being a palindrome will not persist throughout the iteration even if ever attained. We consider here the problem of taking palindromic differences; starting with X\, define x fe + l \ x k ^kl inductively. In this case, if x k were a palindrome, all its successors would vanish, and the first question that arises is whether this always occurs. This problem has been considered previously (see [1], [4], [5]). Clearly, if X\ has only one digit, then x 2-0, and if Xi has two digits, then x 2 will have at most two digits and be divisible by 9. If x 2 = 9 or 99, then # 3 = 0 , whereas all other cases do eventually reach zero, as the sequence 90, 81, 63, 27, 45, 9, 0 shows, for this sequence together with all palindromic inverses contains all integers of no more than two digits divisible by 9. The same reasoning applies to three-digit numbers, for then x 2 will be divisible by 99, and the sequence 990, 891, 693, 297, 495, 99, 0 shows just as before that, for any Xi under 1000, the process leads to zero in the end. As we shall see presently, the close connection between the behavior for two-and three-digit numbers is not mere coincidence. Given an x± having n digits, it is not necessarily true that x 2 < #]_> but certainly x 2 has n or fewer digits. Accordingly, from any starting point Xi of digit length n one of two things …