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Let typeset structure be the integer obtained from the positive integer typeset structure by reversing the digits of typeset structure written to the base typeset structure. The result of the reversal addition  typeset structure is called a versum  (a term introduced by Michael T.Rebmann and Frederick Groat in 1965 [1]  ).  Define typeset structure and  typeset structure.

An old conjecture states that for every positive integer typeset structure there is a typeset structure with a palindromic typeset structure.

This conjecture has been proved false for integers expressed in bases of the form typeset structure.  D.C.Duncan  [2]  and independently R.Sprague [3]  , H.Gabai and D.Cooganm [4] and A.Brousseau  [5]  disproved this conjecture by a counter-example in which the versum sequence typeset structure, typeset structure, exhibits a palindrome-free recursive cycle of four versums.  Brousseau found in addition two other palindrome-free sequences, each with a different palindrome-free recursive cycle of four versums. The  smallest number that never becomes palindromic in the base 2 is 10110 (decimal 22).  

Motivated by a Duncan remark that ... it would be highly interesting to establish the existence of numbers that neither become palindromic nor show a periodic recursion of cycles of digits. Ch.Trigg [1]  showed that there is an unlimited number of distinct palindrome-free recursive cycles in the binary scale.

In the base 4, the number 255 (decimal) leads to a palindrome-free sequence.

The conjecture remains unsolved for any other base. In what follows consider the base typeset structure.

For instance  typeset structure, and the initial segment of its versum sequence is  281, 463, 827, 1555, 7106, 13123, 45254.  Note that in a versum sequence there may appear more palindromes. For instance, 45254 is not the only palindrome in the versum sequence for 281. The sequence continues with  90508, 171017, 881188, ... .

The smallest integer of the set of integers having the same typeset structure is called a basic integer.  The set contains integers having the same middle digits and their symmetrically located digit pairs have the same sums.   For instance, 10552 is a basic integer for the set {10552, 11542, 12532, 15322, 15412, 15502, 20551, 21541, 22531, 23521, 24511,25501} and typeset structure.  There are 404 basic integers typeset structure in the binary scale [1] .

No integer typeset structure that produces a palindrome for typeset structure needs more than 24  versum operations   [6]  .  For instance typeset structure. There are 1515 five-digit integers represented by 61 basic integers which require 24 or more versum operations to produce a palindrome   [7] . For instance typeset structure .

D.H.Lehmer reports (Sphinx 8 (1938), pp.12-13) that no palindrome occurs for typeset structure for  typeset structure, nor for typeset structure, typeset structure.  To check this visit 

196 is the smallest known number for which the above versum procedure has not yet produced a palindrome.

By the year 1972 the computations have shown  typeset structure, typeset structure, typeset structure and typeset structure with no palindrome encountered.  

J.Walker  reports on a 3-years’ spare time calculation on a Sun 3/200, which after 2,415,836 versum operations reached   a number containing 1 million digits, without ever yielding a palindrome.  T.Irvin reports   on a project building on  Walker's efforts  and which  took only two months spare time on a supercomputer to reach a nonpalindromic number of two million digits.

To generate a versum sequence for some other seeds visit  

References

[1]  Trigg, C. W. (1973). Versum sequences in the binary system. Pacific J. Math., 47, 263-275 (corrections, ibid. 49 (1973), 619).

[2]  Duncan, D. C. (1939). Sijet d’étude, No. 74. Sphinx (Bruxelles), 9, 91-92.

[3]  Sprague, R. (1963). Recreation in mathematics.  Problem 5, pages 6-7, 28-29; Dover Publications Inc..

[4]  Gabai, H., & Coogan, D. (1969). On palindromes and palindromic primes. Math. Mag., 42, 252-254.

[5]  Brousseau, A. (1969). Palindromes by addition in base two. Math. Mag., 42, 254-256.

[6]  Trigg, C. W. (1967). Palindromes by addition. Math. Mag., 40, 26-28.

[7]  Rebmann, M. T., & Sentyrz Jr., F. (1972). A note on palindromes by reversal-addition. Math. Mag., 45(4), 186-187.

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Štefan Porubský: 196.

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