25i 



Note on IMultiply Perfect Numbeus. Inci.i'ding a Table of 204 

 New Ones and the 47 Others PKEViorsLY Published. 



By K. D. Cakmiciiael and T. E. Mason. 



§1. I lit rod uci ion and Historical ISlotc. 



If the snin of all the divisors of X is niN, where m is an integer, we 

 bhall tall N a multiply ixn'fect Jiuniber of multiplicity m. If m=2 we shall 

 call N a perfect number. 



The study of such numbers gave rise to tlie principal contributions of 

 F( rmat to the Iiigher arithmetic ; and consequently they have been a means 

 of prime importance in leading to the development of the modern theory 

 of nun.ibers.^ As is well kr.o\A'n their history goes back to Euclid, wlio 

 proved that every number of the form 2"~U2''-1). where 2^—1 is a prime, is 

 a perfect number. Euler and others- have shown that every even perfect 

 raimber is of the Euclid type; but it remains an open question as to 

 v.hether there do or do not exist odd perfect numbers. Several supposed 

 proofs that no otld perfect number exists have been given, but none of 

 liiese is rigorous. The actually known perfect numbers' are included in 

 the Euclid formula 2P-'(23-l) for the ten values of p, p = 2, 3, 5, 7, 13, 17, 

 19, 31, Gl. S9. 



It appears that the first discovery of a multiply perfect number of 

 nuiltiplicity greater than 2 is due to Mersenne, who observed that 120 is 

 one-third of the sum of all its divisors. In response to a problem proposed 

 by Mersenne, Fermat pointed out that G72 has also the property of being 

 equal to one-third the sum of all of its divisors. From time to time other 

 nmltlply perfect numbers have been discovered.* Up to the present time 



^ Cf. Lucas, Thcorie des nomhrcfi, I, p. 376. 



^ A very simple proof of this theorem has recently been given by Dickson, 

 American Mathematical Monthly, vol. 18 (1911), p. 109. See also a proof by Car- 

 miohael, Annals of Mathematics, vol. 8 (1907), p. ].')0. 



^ For reference to the literature of perfect numbers, see Encyclopedic des 

 sciences mathematiques, I3, pp. 53-56. 



■* For a short history of these numbers, with references, see EncyclopMie den 

 sciences mathematiques, I3, pp. 56-58. 



[17—29034] 



