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Four Integers whose Twelve Quotients Sum to Zero

Published online by Cambridge University Press:  20 November 2018

John Leech
John Leech*
Affiliation:
University of Stirling, Stirling, Scotland, U.K
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This paper is devoted to the study of sets of four unequal integers ni such that the twelve quotients ni /nj (i ≠ j) of pairs of distinct members sum to zero. (Without the restriction i ≠ j the condition would be equivalent to (Σ ni )(Σ l/ni ) = 0, of no great interest.) Constructions for these sets are given, and relations between them are studied. It is found that each set belongs to an orbit of six related sets, and that each such orbit is related to four neighbours, each of which is another orbit of six sets. A study is made of the graph formed by assigning a node to each orbit of six solutions and joining it to the nodes assigned to its four neighbours. This appears to comprise one component containing an infinity of cycles together with an infinite forest of infinite trees.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 38 , Issue 5 , 01 October 1986, pp. 1261 - 1280
Copyright
Copyright © Canadian Mathematical Society 1986

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