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 + 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, exce"dant, ubervollstandig, uberflussig, uberschiessende). Euclid1 proved that, if p = l+2+22+ ... +2n is a prime, 2np is a perfect number. He showed that 2np is divisible by 1, 2,..., 2n, p, 2p}..., 2n~lp, but by no further number less than itself. By the usual theorem on geometrical progressions, he showed that the sum of these divisors is 2np. The early Hebrews10 considered 6 to be a perfect number. Philo Judeus16 (first century A. D.) regarded 6 as the most productive of all numbers, being the first perfect number. Nicomachus2 (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 Smyrna3 (about A. D. 130) distinguished between perfect (citing 6, 28), abundant (citing 12) and deficient (citing 8) numbers. 'Elernenta, liber IX, prop. 36. Opera, 2, Leipzig, 1884, 408. lfflS. Rubin, "Sod Hasfiroth" (secrets of numbers), Wien, 1873, 59; citation of the Bible, Kings, II, 13, 19. 1&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. 'Nicomaehi Geni.siui arithrncticae libri duo. Nunc primum typis excusi, in lucem eduntur. Parisus, 1538. In officina Christian! Wecheli. (Greek.) TheoIoKumena arithmetical. Accedit Nicomachi Gerasirii institutio arithmetics ad fidem codieurn MonaceriHiurn emcndata. Ed., Fridericus Astius. Lipsiae, 1817. (Greek.) Nicomachi Gerascni Pythagorci introductions arithmetics libri ii. llecensvit Ricardua Hoche. Lipsiac, 1866. (Greek.) 8Theonis Smyrriaei philosophi Platonici expositio rerurn rnathematicarum ad legendum Platoncm utilium. Ed., Ed. Ililler, Leipzig, 1878, p. 45. Theonis Srnyrnaei PJatonici, Latin by Ismaele Bullialdo. Paris, 1644, chap. 32, pp. 70-72.