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Let be the integer obtained from the positive integer by reversing the digits of written to the base . The result of the reversal addition is called a versum (a term introduced by Michael T.Rebmann and Frederick Groat in 1965 [1] ). Define and .
An old conjecture states that for every positive integer there is a with a palindromic .
This conjecture has been proved false for integers expressed in bases of the form . 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 , , 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 .
For instance , 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 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 . There are 404 basic integers in the binary scale [1] .
No integer that produces a palindrome for needs more than 24 versum operations [6] . For instance . 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 .
D.H.Lehmer reports (Sphinx 8 (1938), pp.12-13) that no palindrome occurs for for , nor for , . 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 , , and 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 .
[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.