✦ LIBER ✦

Common factors of integers: A graphic view

✍ Scribed by R.B Eggleton

Publisher: Elsevier Science
Year: 1987
Tongue: English
Weight: 492 KB
Volume: 65
Category: Article
ISSN: 0012-365X
DOI: 10.1016/0012-365x(87)90136-1

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✦ Synopsis

The common factor graph of a set of integers has the integers as vertices, two vertices being adjacent just if they have a proper common factor. Such graphs permit visual interpretation of many common factor properties of sets of integers. A characterization of common factor graphs is given. The common factor graph of P, the set of integers ~>2, is a diameter 2 graph in which every induced subgraph is a common factor graph, and every common factor graph is isomorphic to an induced subgraph of the common factor graph of P. We discuss the problem of finding the length of the smallest initial segment of P which contains a given finite graph as an induced subgraph.

Connected common factor graphs of runs of consecutive integers are considered in detail. Pillai and Brauer proved that there exist runs of n consecutive integers not containing any member coprime to all the rest, precisely when n ~> 17. A new uniform construction is given for this result. The paper concludes with relevant numerical results, including constellations of runs with connected common factor graphs occurring around 151 058 and 771 320.

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