38 HISTORY OF THE THEORY or NUMBERS. [CHAP. I % three distinct prime factors;329 that those with only four distinct prime factors are330 the P3(3) of St. Croix306 and the P*(1) of Descartes;309 and that the even Pm with five331 distinct prune factors are F3(4), P®\ A(3) of Descartes308' 309 and P4(8) of Mersenne.314 Carmichael3310 stated and J. Westlund proved that if n>4, no Pn has only n distinct prime factors. Carmichaers332 table of multiply perfect numbers contains the misprint 1 for the final digit 0 of Descartes' P4(3), and the erroneous entry 919636480 in place of its half, viz., P3(5) of Mersenne.314 The only new Pm is P6(7)=21535527211.13.17-19.3143-257. All Pm<10Q were determined; only known ones were found. Carmichael333 gave an erroneous P5 and the new P4: P4<") =214327213-1923M27-151, P4(16>=22536549.23437.547.6834093-2731.8191. Carmichael and T. E. Mason334 gave a table which includes the above listed 10 P2j 6 P3, 16 P4, 8 P5, 7 P6, 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 Mh powers of whose divisors <n is a multiple of n. For example, n = 2py where p is a prime 8/1=*= 3 and k is such that 2k-\-l is divisible by p'} cases are p=3, fc = l; p = 5, fc=2; p = ll, /b=5; p = 13, fc=6.] AMICABLE NUMBERS. Two numbers are called amicable* if each equals the sum of the aliquot divisors of the other. According to lamblichus4 (pp. 47-48), " certain men steeped in mistaken opinion thought that the perfect number was called love by the Pythagoreans on account of the union of different elements and affinity which exists in it; for they call certain other numbers, on the contrary, amicable numbers, adopting virtues and social qualities 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. When asked what is a friend, he replied, 'another I/ which is shown in these numbers. Aristotle so defined a friend in his Ethics." '"Annals of Math., (2), 7, 1905-6, 153; 8, 1906-7, 49-56; 9, 1907-8, 180, for a simpler proof that there ia no J°8 = piap2&P»c, c> 1. ""Annals of Math., (2), 8, 1906-7, 149-158. MlBull. Amer. Math. Soc., 15, 1908-9, pp. 7-8. Fr. transl., Sphinx-Oedipe, Nancy, 5, 1910, 164-5. "^Amer. Math. Monthly, 13, 1906, 165. «2Bull. Amer. Math. Soc., 13, 1906-7, 383-6. Fr. transl, Sphinx-Oedipe, Nancy, 5, 1910, 161-4. «3Sphinx-Oedipe, Nancy, 5, 1910, 166. «<Proc. Indiana Acad. Sc., 1911, 257-270. *Amiable, agreeable, befreundete, verwandte.