260 



IS a mulliphj perfect number of multiplicUy mj'. 



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



3^ 7 . IP. 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 only to show in 

 each case that 



fli-l-l Pi+i 



mi «i+i ni; ft' 



Pi(pi-i) qi^qi-i) 



The verification is omitted. 



The following theorem, due to De.scartes, i.s 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 ?nultiply perfect 



number of multiplicity (p + 1)'^ . 



%?,. Table of Miiltiph) Perfect A'»?/)/>cr/5.*t 



2) 2. o. (Euclid, Nicomaque.) 



2) 2-. 7. (Euclid, Nicomaque.) 

 4) 2-. 31 5. 7-. 13. 19. (Lehmer.) 



3) 2\ 3. 5. (Mersenne.) 



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



2) 2\ 31. (Euclid, Nicomaque.) 



3) 2\ 3. 7. (Fermat.) 



4) 2\ 3\ 5. 7. (Descartes.) 

 4) 2'. 3\ 7-. 11'. 19=. 127. 



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



4) 2^ 3\ 5^ 17. 31. (Mersenne.) 



5) 2^ 3'. 5. 7. ir. 17. 19. (Descartes.) 

 5) 2\ S\ 5. 71 13. 17. 19. (Descartes.) 



4) 2\ 3". 5. 17. 23. 137. 547. 1093. (Fermat.) 



4) 2'. S'". 5. 17. 23. 107. 3851. 



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



• The numbers marked with a star were discovered by Mr. Mason. The re- 

 maining hitherto unpublished numbers wore discovered by Mr. Carmieliael. 



t The multiplicity of each number Is written to its left. If pre\iously (lub- 

 lislii'd llie discoverer's name is Kiv(>n to Ihe rijiht. 



