Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 36 
In the ninth century the Arab Thabit ben Korrah^° (prop. 10) noted that 
2'*/i^ and 2"s are amicable numbers if 
(1) /i = 3-2"-l, f=3-2"-i-l, s = 9'2^''-'-l 
are primes > 2, literally, if 
/l = 2+2^ t = z-2''-\ 3 = H-2+...+2^ s = (2"+^+2'*-2)2''+^-l. 
The term used for amicable numbers was se invicem amantes. In the article 
in which F. Woepcke^'^ translated this Arabic manuscript into French, he 
noted that a definition of these numbers, called congeneres, occurs in the 
51st treatise (on arithmetic) of Ikhovan Algafa, manuscript 1105, anciens 
fonds arabes, p. 15, of the National Library of Paris. 
Among Jacob's presents to Esau were 200 she-goats and 20 he-goats, 
200 ewes and 20 rams (Genesis, XXXII, 14). Abraham Azulai^^^ (1570- 
1643), in commenting on this passage from the Bible, remarked that he had 
found written in the name of Rau Nachshon (ninth century A. D.): Our 
ancestor Jacob prepared his present in a wise way. This number 220 (of 
goats) is a hidden secret, being one of a pair of numbers such that the parts 
of it are equal to the other one 284, and conversely. And Jacob had this in 
mind; this has been tried by the ancients in securing the love of kings and 
dignatories. 
Ibn Khaldoun^^° related "that persons who have concerned themselves 
with talismans affirm that the amicable numbers 220 and 284 have an 
influence to establish a union or close friendship between two individuals. 
To this end a theme is prepared for each individual, one during the ascend- 
ency of Venus, when that planet is in its exaltation and presents to the 
moon an aspect of love or benevolence; for the second theme the ascendency 
should be in the seventh. On each of these themes is written one of the 
specified numbers, the greater (or that with the greater sum of its aliquot 
parts?) being attributed to the person whose friendship is sought." 
The Arab El Madschriti,^^! or el-Magriti, (flOOT) of Madrid related that 
he had himself put to the test the erotic effect of ''giving any one the 
smaller number 220 to eat, and himself eating the larger number 284." 
Ibn el-Hasan^"'' (tl320) wrote several works, including the "Memory 
of Friends," on the explanation of amicable numbers. 
Ben Kalonymos^^^^ discussed amicable numbers in 1320 in a work 
written for Robert of Anjou, a fragment of which is in Munich (Hebr. MS. 
290, f. 60). A knowledge of amicable numbers was considered necessary 
by Jochanan Allemanno (fifteenth century) to determine whether an 
aspect of the planets was friendly or not. 
^^^Baale Brith Abraham [Commentary on the Bible], Wilna, 1873, 22. Quotation suppUed by 
Mr. Ginsburg. 
'^oProlegomenes hist. d'Ibn Khaldoun, French transl. by De Slane, Notices et Extraits des 
Manuscrits de la Bibl. Imperiale, Paris, 21, I, 1868, 178-9. 
"^'Manuscript Magriti; Steinschneider, Zur pseudoepigraphischen Literatur inbesondere der 
geheimen Wissenschaften des Mittelalters, Berlin, 1862, p. 37 (cf. p. 41). 
'""H. Suter, Abh. Gesch. Math. Wiss., 10, 1900, 159, § 389. 
»"*Hebr. Bibl., VII, 91. Steinschneider, Zeitschrift der Morgenlandischen Ges., 24, 1870, 369. 
