CHAPTER I. 
PERFECT. MULTIPLY PERFECT. AND AMICABLE NUMBERS. 
Perfect, Abundant, and Deficient Numbers. 
By the aliquot parts or divisors of a number are meant the divisors, 
including unity, which are less than the number. A number, like 6 = 1 -h 
2+3, which equals the sum of its aliquot divisors is called perfect (voll- 
kommen, vollstandig) . If the sum of the aliquot divisors is less than the 
number, as is the case with 8, the number is called deficient (diminute, 
defective, unvollkommen, unvollstandig, mangelhaft). If the sum of the 
aliquot divisors exceeds the number, as is the case with 12, the number is 
called abundant (superfluos, plus quam-perfectus, redundantem, exc^dant, 
iibervollstandig, iiberflussig, iiberschiessende) . 
Euclid^ proved that, if p = 1+2+2^+ • • • +2" is a prime, 2"p is a perfect 
number. He showed that 2"p is divisible by 1, 2, . . . , 2", p, 2p, . . . , 2'*~^p, 
but by no further number less than itself. By the usual theorem on 
geometrical progressions, he showed that the sum of these divisors is 2"^. 
The early Hebrews^" considered 6 to be a perfect number. 
Philo Judeus^'' (first century A. D.) regarded 6 as the most productive 
of all numbers, being the first perfect number. 
Nicomachus^ (about A. D. 100) separated the even numbers (book I, 
chaps. 14, 15) into abundant (citing 12, 24), deficient (citing 8, 14), and 
perfect, and dwelled on the ethical import of the three types. The perfect 
(I, 16) are between excess and deficiency, as consonant sound between 
acuter and graver sounds. Perfect numbers will be found few and arranged 
with fitting order; 6, 28, 496, 8128 are the only perfect numbers in the 
respective intervals between 1, 10, 100, 1000, 10000, and they have the 
property of ending alternately in 6 and 8. He stated that Euclid's rule 
gives all the perfect numbers without exception. 
Theon of Smyrna^ (about A. D. 130) distinguished between perfect 
(citing 6, 28), abundant (citing 12) and deficient (citing 8) numbers. 
^Elementa, liber IX, prop. 36. Opera, 2, Leipzig, 1884, 408. 
^"S. Rubin, "Sod Hasfiroth" (secrets of numbers), Wien, 1873, 59; citation of the Bible, 
Kings, II, 13, 19. 
**Treatise on the account of the creation of the world as given by Moses, C. D. Young's 
transl. of Philo's works, London, 1854, vol. 1, p. 3. 
'Nicomachi Gerasini arithmeticse Ubri duo. Nunc primdm typis excusi, in lucem eduntur. 
Parisiis, 1538. In officina Christian! WecheU. (Greek.) 
Theologumena arithmeticae. Accedit Nicomachi Gerasini institutio arithmetica ad fidem 
codicum Monacensium emendata. Ed., Fridericus Astius. Lipsiae, 1817. (Greek.) 
Nicomachi Geraseni Pythagorei introductionis arithmeticae libri ii. Recensvit Ricardus 
Hoche. Lipsiae, 1866. (Greek.) 
'Theonis Smymaei philosophi Platonici expositio rerum mathematicarum ad legendum 
Platonem utiHum. Ed., Ed. Hiller, Leipzig, 1878, p. 45. 
Theonis Smymaei Platonici, Latin by Ismaele BuUialdo. Paris, 1644, chap. 32, pp. 70-72. 
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