190 MATHEMATICSMAGAZINE Upper Bounds on the Sum of PrincipalDivisors of an Integer ROGER B. EGGLETON Illinois State University Normal, IL61790
[email protected] WILLIAM P. GALVIN School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308 Australia A prime-poweris any integerof the form p", where p is a primeand a is a positive integer.Two prime-powersare independentif they are powers of differentprimes. The Fundamental Theorem of Arithmetic amounts to the assertion that every positive integerN is the productof a uniqueset of independentprime-powers, which we call theprincipal divisors of N. Forexample, 3 and4 arethe principaldivisors of 12, while 2, 5 and 9 are the principaldivisors of 90. The case N = 1 fits this description,using the conventionthat an emptyset has productequal to 1 (andsum equalto 0). In a recentinvited lecture, Brian Alspach noted [1]:Any odd integerN > 15 thatis not a prime-power is greater than twice the sum of its principal divisors. For instance, 21 is morethan twice 3 plus 7, and 35 is almostthree times 5 plus 7, but 15 falls just shortof twice 3 plus 5. Alspachasked for a "nice"(elegant and satisfying) proof of this observation,which he used in his lectureto provea resultabout cyclic decomposition of graphs. Respondingto the challenge,we proveAlspach's observation by a veryelementary argument.You, the reader,must be the judge of whetherour proof qualifiesas "nice." We also show how the same line of reasoningleads to several strongeryet equally elegantupper bounds on sums of principaldivisors. Perhaps surprisingly, our meth- ods will not focus on propertiesof integers.Rather, we considerproperties of finite sequencesof positivereal numbers,and use a classicalelementary inequality between the productand sum of any such sequence.But first,let us put Alspach'sobservation in its numbertheoretic context.