258 



there have been published altogether, so far as we have been alile to find 

 out. a total of forty-seven multiply perfect numbers. Cunningham^ has 

 announced that he has a table of eighty-tive multiply perfect numbers; 

 but lie published only one of them. In the table of perfect and multiply 

 lierfect numbers in §3 we have credited to the discoverer each of the forty- 

 sincn numbers which have heretofore been published.- The remaining two 

 hundred four luunbers of the table are believed to be publislied here for 

 the tirst time. It should be noted that numbers of multiplicity 7 occur in 

 this table for the tirst time. 



In §2 we have given some working rules which. were found useful in 

 obtaining new multiply perfect numbers from those already known or dis- 

 covered in the process of constructing the table. Their further use would 

 consist in the possible discovery of several new nniltiply perfiH't numbers 

 from a single new one found by any other means whatever. It was in 

 this way that many of the new numl>ers in this pai»er were discovered; 

 one was obtaine<l by direct means and others followed by use of the rules. 

 As to the rules tliemselves, some of them were gotten by direc-t means and 

 others by comparison of numbers in the tal)le while the table it.self was 

 being con.structed. The list of nuiiihcr pairs in tlic rules might be largely 

 extende<l by a further comparison of mnnbers in the table. We have 

 selected a i)art of tliose which actually proveil to be of most use iu the 

 construction of tlie tal)Ie. 



S2. Ritli'S for FiiKliiii/ Miiltii)li/ Perfect Xiniihrrs. 

 The following two tlieorems afford useful working rules for finding 

 new multiitly perfect numbers : 



I. //"II Pi"' and 11 q-,/''' (in either order) are a pair of factor sets from the list 

 below and if a multiply perfect number N of multiplicity m contains the factor 

 II jV'i without containing either any fador Pi"'+ ' or any factor q, different from 

 every pi) then the number 



N n qjh 



n pi«i 



is also a multiply perfect number of multiplicity m. 



^British Anfuxiatiiin I{<t)<,r1. X'MrJ.. pp. .528-529. 



2 We are indibtcd to I'rcf. Dickson for leforence to this first publication of six 

 of these numbers, 



