251 



Note on Multiply Perfect Nitmheiis. Including a Table of 204 

 New Ones and the 47 Others Previottsly Published. 



i;y 11. I). Cakjiuhael and T. E. Mason. 



§1. Intr<j(liirH()ii, and Historical Note. 



If the sum of all the divisors of N is niN, where iii is fiu integer, we 

 irhall call N a multiply ijorfect ]uiml)er of uuiltii)li(ity m. If m=2 we shall 

 call N a perfect number. 



The study of such numbers gave rise to the priueiiial contributions of 

 Fcrmat to the higher arithmetic; and conseciuently they have been a means 

 of prime importance in leading to the development of the modern theory 

 of numbers.^ As is well known their history goes back to Euclid, who 

 i;roved that every number of the form 2'"'(2''-l). where 2'-l is a prime, is 

 u perfect number. Euler and others^ have shown that every even perfect 

 number is of the Euclid type; but it remains an open question as to 

 v.hether there do or do not exist odd perfect numbers. Several supposed 

 proofs that no odd perfect number exists have been given, but none of 

 these is rigorous. The actually known i)erfect numbers- are included in 

 the Euclid formula 2P-'(2P-1) for the ten values of p, p = 2, 3, 5, 7, 13, 17, 

 10, 31, 61, SO. 



It appears that the first discovery of a multiply perfect number of 

 multiplicity greater than 2 is due to Mersenne, who observed that 120 is 

 one-third of the sum of all its divisors. In response to a problem proposed 

 by Mersenne, Fermat pointed out that G72 has also the property of being 

 equal to one-third the sum of all of its divisors. From time to time other 

 multiply perfect numbers have been discovered.* Up to the present time 



1 Cf. Lucas, Theorie des nom'brcs, I, p. 376. 



2 A very simple proof of this theorem has recently been given by Dickson, 

 American Mathematical Monthly, vol. 18 (1911), p. 109. See also a proof by Car- 

 michael, Annals of Mathematics, vol. 8 (1907), p. 3 50. 



3 For reference to the literature of perfect numl)ers, see Encifclopcdie des 

 sciences mathematiques, I3, pp. .53-56. 



^ For a short history of these numbcis, with references, see Encyclopedic den 

 sciences mathematiques, I3, pp. 56-58. 



[17—29034] 



