From: Gerry Myerson Subject: Re: Palindromic Sum Conjecture Date: Thu, 16 Mar 2000 14:16:22 +1100 Newsgroups: sci.math Summary: [missing] In article <8ao3jo$usg$1@bunyip.cc.uq.edu.au>, "Brett W" wrote: > My favourite unsolved problem that I like toying with is the Palindromic > Sum Conjecture (I incidentally don't expect to solve it, but it seems very > interesting). > > The Palindromic Sum Conjecture goes as follows: > Given any number, n, add to it its reverse (i.e. 123 becomes 321). > Eventually every number produces a palindrome by this method. > > Has anyone solved it yet? What headway has been made on it? I believe > that in binary it is possible to find numbers that never yield palindromes. Here's what David Wells says in The Penguin Dictionary of Curious and Interesting Numbers: 196 is the only number less than 10000 that by this process has not yet produced a palindrome.... P. Anderton has taken this up to 70928 digits.... In base 2, it is certainly not true that every number eventually generates a palindrome. Roland Sprague shows that 10110 never does so. Wells gives no references. The book was published in 1986. A search for "palindrome" at Math Reviews didn't turn up any more recent papers that struck me as relevant. Some older papers follow. My apologies for the formatting. Gerry Myerson (gerry@mpce.mq.edu.au) ************************************ 50 #12891 10A30 Alter, Ronald; Curtz, Thaddeus B. Remarks and results on palindromes. Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), pp. 349--359. Congressus Numerantium, No. X, Utilitas Math., Winnipeg, Man., 1974. An integer $N$ written in base $B$ is called a palindrome if $N$ has the same sequence of digits reading from left to right and right to left. If $N=S\sb 0$, define $S\sb {k+1}=S\sb k+S\sb k{}'$, where $S\sb k{}'$ denotes the number obtained by writing $S\sb k$ in reverse order. The authors prove that if $S\sb k$ is a palindrome then (i) $(B+1)\vert S\sb k$ if $S\sb k$ has an even number of digits, (ii) $(B+1)\sp 2\vert S\sb k$ if $S\sb k$ has an odd number of digits and some previous $S\sb j$ has an even number of digits, (iii) in the addition $S\sb {k-1}+S\sb {k-1}'$ either no carry occurs, or there is a carry in the last position. If $N$ is a two-digit number in base $B$, conditions are given that imply that some $S\sb k$ is a palindrome. Finally, a table is given of the smallest value $N$ in each base $B\leq 51$, for which no $S\sb k$, $k\leq 100$, is a palindrome. \{For the entire collection, see MR 50 #4315.\} Reviewed by W. A. Webb 48 #5993 10A40 Trigg, Charles W. Versum sequences in the binary system. Pacific J. Math. 47 (1973), 263--275; corrections, ibid. 49 (1973), 619. The result $S'$ of adding a positive integer $S$, relative to some base, to the integer whose digits are those of $S$ reversed, is called a versum. An old conjecture is that, relative to base 10, any iterative sequence of versa $S,S',S",S"',\cdots$ contains a palindromic number. D. C. Duncan [Sphinx 9 (1939), 91--92] showed that the conjecture is false relative to base 2 by exhibiting a palindromefree recursive cycle of 4 versums. The present author, also working in base 2, is concerned with further instances of such palindrome-free sequences as well as with palindromefree sequences not exhibiting such recursive cycles. Reviewed by J. B. Roberts 40 #7193 10.09 Brousseau, Brother Alfred Palindromes by addition in base two. Math. Mag. 42 1969 254--256. The author has examined the binary representations of all integers $N\leq 650$ with a view to producing further instances of the failure of the conjecture discussed in the preceding review [\#7192 above]. The number 837 is really $S\sb 9$ for $N=22$. Two other types of failure are noted corresponding to $N=77$ and $775$. In cases where the conjecture holds, $k\leq 11$ for all $N\leq 650$. Reviewed by D. H. Lehmer