Vol. 47, No. 1, 1973

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Versum sequences in the binary system

Charles W. Trigg

Vol. 47 (1973), No. 1, 263–275

Let the reverse of any positive integer N be Nand N + NJ = S1. Then

S1 + S′1 = S2,S2 + S2′= Ss, , Sk −1 + S′k−1 = Sk.

Each of the Si, the result of a reversal-addition operation on an integer, can appropriately be called a versum (a term coined in 1965 by Michael T. Rebmann and Frederick Groat while undergraduates at Carleton College). Thus, reiteration of the operation generates a sequence of versums. An ancient conjecture says that for every N there is a palindromic Sk. True for one-digit and two-digit N’s in the decimal system, considerable doubt has been thrown upon the universal verity of the conjecture [1, 2].

In the binary system, D. C. Duncan disproved the conjecture by a counter-example in which the sequence exhibits a palindrome-free recursive cycle of four versums (hereafter referred to as a PFRC-4). Sprague, Gabai and Coogan, and Brousseau rediscovered the same cycle. Brousseau also found two other palindrome-free sequences, each with a different PFRC-4.

Duncan remarked 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.” Such integers are reported below, together with an unlimited number of distinct palindrome-free recursive cycles in the binary system.

Mathematical Subject Classification
Primary: 10A40
Received: 16 May 1972
Published: 1 July 1973
Charles W. Trigg