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Upper Bounds on the Sum of Principal Divisors of an Integer

Award: Carl Allendoerfer

Year of Award: 2005

Publication Information: Mathematics Magazine, Vol. 77 (2004), pp. 190-200

Summary: This article gives an elementary argument proving the statement, "Any odd integer N > 15 that is not a prime-power is greater than twice the sum of its principal divisors."

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About the Authors

Roger B. Eggleton is an Australian-born mathematician whose main fields of interest are combinatorics, graph theory and number theory.  His research and teaching career encompasses universities in five countries – Australia (1963-70), Canada (1970-73), Israel (1973-74), USA (1974-76), Australia (1976-88), Brunei (1989-92), and USA (1993-present).  He obtained his Ph.D. in Calgary (1973), under Richard Guy.  He has published over 60 research papers, and regards his collaborations with many joint authors, including several papers with Paul Erdös, as one of the main pleasures of his career.  He published four joint papers with William Galvin (2000-04).

William P. Galvin was born in Sydney, Australia on February 5, 1938.  His professional career began as a high school mathematics and science teacher, transforming by 1970 into teacher training.  Always a dedicated student, he completed four degrees while working full-time: B.A. (Sydney, 1962), M. Ed. (Newcastle, 1974), M. Math. (Newcastle, 1977), M. Eng. Sc. (Newcastle, 1982), all three masters degrees being research degrees. By 1989 Bill was head of the department responsible for teaching mathematics, computing and mathematics education at the Hunter Institute of Higher Education – subsequently amalgamated with University of Newcastle, where Bill continued training mathematics teachers until he retired in 1997.  After retirement his research and other mathematical involvements continued unabated, despite growing ill health.  Soon after completing a three-year term as coeditor of the Australian Mathematical Society’s Gazette (2001-03), he died of cancer on December 12, 2003.  The next issue fittingly published his obituary, see Australian Mathematical Society’s Gazette, 31 (2004), pp. 4-5.

Author (old format): 
Roger B. Eggleton, William P. Galvin
Author(s): 
Roger B. Eggleton, William P. Galvin
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Publication Date: 
Wednesday, October 15, 2008
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Summary: 

This article gives an elementary argument proving the statement, "Any odd integer N > 15 that is not a prime-power is greater than twice the sum of its principal divisors"

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