259 



2M7, 2'°. 23. 89 



2'. 17, 2". 19.683.2731.8191 



2^31, 21^43. 127t 



211.35^ 2-«. 33.127. 337 



2-\ 7 . 23 . 233 . 1103 . 2089, 2'\ 7K 43 . 223 . 7019 . 112303 . 898423 . 61G318177 



2". 131071, 2". 174763 . 524289 



238. 53 229 . 8191 . 121369, 2^\ 59. 157 . 43331 . 3033169 . 715827883 . 2147483647 



3MP.13, 3MI.I32 



3M37 . 547 . 1093, 3'°. 107 . 3851 



3^23.41, 3'°. 23=. 79. 107.3851 



5'-.7M9.31, 53.73.13 



5M3-.31-.61.83.331, 53.133.17 



5^7M9=. 127, 5^73.19 



53. 1\ 133. n\ .307 . 467 . 2801, 5^ 73. 13 .17.71 



If N = ri/i r2/2 Tn/" , where ri, r-,. . . ., r„ are different primes, is a 



multiply perfect number of multiplicity m, then from the formula for the 

 sum of all the divisors of N and the fact that this sum is now supposed to 

 be mN, we have 



n T;— 1 



m = II 



i = 1 > i 



ri(ri-l) 



Therefore in order to prove the accuracy of the rules we have only to show 

 in each case that 



«i + 1 /5i + 1 



Pi - 1 q; - 1 



Pi (Pi-1) Qi (Qi-l) 



The verification is not carried out. 



II. If II Pi"i (mi) and H Qj ji\ (mo) {in either order) are a pair of factor sets 

 and multiplicity from the list below and if a multiply perfect number Ni of mul- 

 tiplicity nix contains thz factor n p;«i without containing either any factor p/''+l 

 or any factor cj; different from cv;ry pi; then the number 



N,nq,,?i 



N. 



IT pi«i 



This pail' is due to Descartes. 



