38 History of the Theory of Numbers. [Chap, i 
three distinct prime factors ;^^^ that those with only four distinct prime fac- 
tors are^^" the P^^^^ of St. Croix^"^ and the P4^'^ of Descartes f°^ and that the 
even P„ with five^^^ distinct prime factors are P3^^\ Pi^\ Pi^^ of Des- 
cartes^"^' ^"^ and P^^^^ of Mersenne.^" 
CarmichaeP^^'* stated and J. Westlund proved that if n>4, no P„ has 
only n distinct prime factors. 
Carmichael's^^^ table of multiply perfect numbers contains the misprint 
1 for the final digit of Descartes' Pi^\ and the erroneous entry 919636480 
in place of its half, viz., P^-^^ of Mersenne.^^^ The only new P^ is 
Pg(7) = 2^^3^527211. 13-17.19-3143.257. 
All P,;,< 10^ were determined; only known ones were found. 
CarmichaeP^^ gave an erroneous P5 and the new P^: 
p^(i4)^2"3272l3-1923M27.151, 
p^(i5)^2253^52l9-31.683.2731-8191, 
p^(i6)^225365-19-23-137-547-683-1093-2731-8191. 
Carmichael and T. E. Mason^^ gave a table which includes the above 
hsted 10 P2, 6 P3, 16 P4, 8 P5, 7 Pq, together with 204 new multiply perfect 
numbers P, (i = 3, . . . , 7) . Of the latter, 29 are of multiplicity 7, each 
having a very large number of prime factors. No P7 had been previously 
published. 
[As a generalization, consider numbers n the sum of the kth. powers of 
whose divisors < n is a multiple of n. For example, n = 2p, where p is a 
prime 8/1 ±3 and k is such that 2*^+1 is divisible by p; cases are p = 3, 
k = l; p = 5, k = 2; p = ll, k = 5; p = 13, A; = 6.] 
Amicable Numbers. 
Two numbers are called amicable* if each equals the sum of the aliquot 
divisors of the other. 
According to lamblichus^ (pp. 47-48), "certain men steeped in mistaken 
opinion thought that the perfect number was called love by the Pythago- 
reans on account of the union of different elements and affinity which exists 
in it; for they call certain other numbers, on the contrary, amicable num- 
bers, adopting virtues and social quahties to numbers, as 284 and 220, for 
the parts of each have the power to generate the other, according to the rule 
of friendship, as Pythagoras affirmed. WTien asked what is a friend, he 
replied, 'another I,' which is shown in these numbers. Aristotle so defined 
a friend in his Ethics." 
«»Aimalsof Math., (2), 7, 1905-6, 153; 8, 1906-7, 49-56; 9, 1907-8, 180, for a simpler proof that 
there is no Pa = Pi^p^Vi^t c> 1. 
""Annals of Math., (2), 8, 1906-7, 149-158. 
"'Bull. Amer. Math. Soc, 15, 1908-9, pp. 7-8. Fr. transl., Sphinx-Oedipe, Nancy, 5, 1910, 164-5. 
wi^Amer. Math. Monthly, 13, 1906, 165. 
»«Bull. Amer. Math. Soc, 13, 1906-7, 383-6. Fr. transl., Sphinx-Oedipe, Nancy, 5, 1910, 161-4. 
»"Sphinx-Oedipe, Nancy, 5, 1910, 166. 
»"Proc. Indiana Acad. Sc, 1911, 257-270. 
*Amiable, agreeable, befreundete, verwandte. 
