Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 5 
Thabit ben Korrah,^° in a manuscript composed the last half of the 
ninth century, attributed to Pythagoras and his school the employment of 
perfect and amicable numbers in illustration of their philosophy. Let 
s = 1+2+ ... +2". Then (prop. 5), 2'*s is a perfect number if s is a prime; 
2"p is abundant if p is a prime <s, deficient if p is a prime >s, and the 
excess or deficiency of the sum of all the divisors over the number equals 
the difference of s and p. Let (prop. 6) p' and p" be distinct primes >2; 
the sum of the divisors <N oi N = p'p"2" is 
a = (2"+i-l)(l+p'+p") + (2"-l)py. 
Hence N is abundant or deficient according as 
a-iV=(2"+^-l)(l+p'+p")-py>0or <0. 
Hrotsvitha,^^ a nun in Saxony, in the second half of the tenth century, 
mentioned the perfect numbers 6, 28, 496, 8128. 
Abraham Ibn Ezra^^" (tll67), in his commentary to the Pentateuch, 
Ex. 3, 15, stated that there is only one perfect number between any two 
successive powers of 10. 
Rabbi Josef b. Jehuda Ankin^^'', at the end of the twelfth century, recom- 
mended the study of perfect numbers in the program of education laid out 
in his book "Healing of Souls." 
Jordanus Nemorarius^^ (tl236) stated (in Book VII, props. 55, 56) that 
every multiple of a perfect or abundant number is abundant, and every 
divisor of a perfect number is deficient. He attempted to prove (VII, 57) 
the erroneous statement that all abundant numbers are even. 
Leonardo Pisano, or Fibonacci, cited in his Liber Abbaci^^ of 1202, 
revised about 1228, the perfect numbers 
1 2^(2^-1) =6, i 2^(2^-1) =28, | 2^(2^-1) =496, 
excluding the exponent 4 since 2^ — 1 is not prime. He stated that by pro- 
ceeding so, you can find an infinitude of perfect numbers. 
i^Manuscript 952, 2, Suppl. Arabe, Bibliotheque imperiale, Paris. Textual transl., except 
of the proofs which are given in modem algebraic notation as foot-notes [as numbers 
were represented by line, in the manuscript], by Franz Woepcke, Journal Asiatique, 
(4), 20, 1852, 420-9. 
"See Ch. Magnin, Theatre de Hrotsvitha, Paris, 1845. 
""Mikrooth Gedoloth, Warsaw, 1874 ("Large Bible" in Hebrew). Samuel Ben Sdadias Ibn 
Motot; a Spaniard, wrote in 1370 a commentary on Ibn Ezra's commentary, Perush ai 
Perush Ibn Ezra, Venice, 1554, p. 19, noting the perfect numbers 6, 28, 496, 8128, and 
citing EucUd's rule. Steinschneider, in his book on Ibn Ezra, Abh. Geschichte Math. 
Wiss., 1880, p. 92, stated that Ibn Ezra gave a rule for finding all perfect numbers. 
As this rule is not given in the Mikrooth Gedoloth of 1874, Mr. Ginsburg of Columbia 
University infers the existence of a fuller version of Ibn Ezra's commentary. 
"^Quoted by Giideman, Das Jiidische Unterrichtswesen wahrend der Spanish Arabischen 
Periode, Wien, 1873. 
*^In hoc opere contenta. Arithmetica decern libris demonstrata .... Epitome i libros 
arithmeticos diui Seuerini Boetij . . . , Paris, 1496, 1503, etc. It contains Jordanus' 
"Elementa arithmetica decern libris, demonstrationibus Jacobi Fabri Stapulensis," and 
"Jacobi Fabri Stapulenais epitome in duos Hbros arithmeticos diui Seuerini Boetij." 
i^Il Liber Abbaci di Leonardo Pisano. Roma, 1857, p. 283 (Scritti, vol. 1). 
