260 



is a multiply perfect number of muUipliciiy m..: 



3^ 5 . 1\ 13 (5) , S^". 7 . 23 . 107 . 3851 (4) 



V.7 .n\ 19 (5), 3«. 23 . 137 . 547 . 1093 (4) 



5.7 (5), 5^7% 13.19 (6) 



5-. 31 (5), 5'. 7. 13 (6) 



In order to prove the theorem it is clear that we have onlj' to show in 

 each case that 



— n Pi~^ = J- n ^' ~ ^ 



mi °'4-l mi /^i 



Pi(Pi-l) Qi^qi-l) 



The verification is omitted. 



The following theorem, due to Descartes, is also readily proved: 



III. If N is a multiply perfect number of multiplicity p', where p is a 



prime number, and if N is not divisible by p, then pN is a multiply perfect 



number of multiplicity (p + 1)'^ . 



§3. Tabic of Multiply Perfect \ii)nherK.''y 



2) 2. ."}. (Euclid, >>'ic()maquo.) 



2) 2^ 7. (Euclid, Nicon)aque.) 

 4) 2". 31 5. 7\ 13. 19. (Lelimer.) 



3) 2\ 3. 5. (Mersemie.) 



4) 2\ 31 5. 7. 13. (Descartes.) 



2) 2'. 31. (Euclid, Niconiaque.) 



3) 2\ 3. 7. (Feruiat.) 



4) 2'. 3^ 5. 7. (Descartes.) 

 4) 2'. 3*. 7^ 11=. 19=. 127. 



2) 2". 127. (Euclid, Nicomaque.) 



4) 2\ 3\ 5=. 17. 31. (Morscnne.) 



5) 2\ 3*. 5. 7. ir. 17. 19. (Descartes.) 

 5) 2'. 3'. 5. 71 13. 17. 19. (Descartes.) 



4) 2\ 3\ 5. 17. 23. 137. 547. 1093. (l-\Tmat.) 



4) 2'. 3'". 5. 17. 23. 107. 3S.51. 



4) 2\ 3. 5. 7. 19. 37. 73. (Lucas.) 



* The numbers markotl with a star wero discovered liy Mr. Mason. The re- 

 maining hitherto unpublished numbers were discovered by IMr. Oarmichaol. 



t The multiplicity of each number Is written to its left, if previously pub- 

 lished the discoverer's name is given to the right. 



