258 



there have been luiblislied altoj,'ethei-, so far as we Iiavt; l)eeii able to liiid 

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

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

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

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

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

 hundred four numbers of the table are believed to be published here for 

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

 this table for the first 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 multiply perfect numbers 

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

 this way that many of the new numbers in this pai>er were discovere<l; 

 one was obtainetl by direct means and others followed by use of the rules. 

 As to the rules themselves, some of them were gotten by direct means and 

 others by comparison of numbers in the table while th(> table itself was 

 being constructed. The list of number pairs in the rules might be largely 

 extended by a further comparison of numbers in the table. We have 

 selected a part of those which actually proved to be of most use in the 

 construction of the table. 



§2. Rules for Fhidhni Mnltiiili/ Perfect yionlK-rs. 

 The following two theorems afford useful working rules for tinding 

 new multiply perfect numbers: 



I. // li i)i"i and IT qj^i (in either order) are a pair of factor sets from the list 

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

 III);"' without containing either ani/ factor \yi"'^+^ or any factor q, different from 

 every pi) then the number 



N II qj/'^i 

 II pi"i 



is also a m,ultiply perfect number of nniltijilicity m. 



^British Associalion Report, 1002, pp. 528-r)2!t. 



" We are indcbtod to I'rof. Dickson for rofcrence to the fust pulilioatiou of six 

 of thope numljer^. 



