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CHAP. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS, 33 G. Wertheim, Anfangsgrunde der Zahlentheorie, 1902. G. Giraud, Periodico di Mat., 21,1906, 124-9. F. Ferrari, Suppl. al Periodico di Mat., 11, 1908, 36-8, 53, 75-6 (Cipolla). P. Bachmann, Niedere Zahlentheorie, II, 1910, 97-101. A. Aubry, Assoc. frang. avanc. sc., 40, 1911, 53-4; 42, 1913; Penseignement math., 1911, 399; 1913, 215-6, 223. *M. Kiseljak, Beitrage zur Theorie der vollkommenen Zahlen, Progr. Agram, 1911. *J. Vaes, Wiskundig Tijdschrift, 8, 1911, 31, 173; 9, 1912, 120, 187. J. Fitz-Patrick, Exercices Math., ed. 3, 1914, 55-7. MULTIPLY PERFECT NUMBERS. A multiply perfect or pluperfect number n is one the sum of whose divisors, including n and 1, is a multiple of n. If the sum is mn, m is called the multiplicity of n. For brevity, a multiply perfect number of multiplicity m shall be designated by Pm. Thus an ordinary perfect number is a P2. Although Robert Recorde39 in 1557 cited 120 as an abundant number, since the sum of its parts is 240, such numbers were first given names and investigated by French writers in the seventeenth century. As a P3 equals one-half of the sum of its aliquot divisors or parts (divisors <Pa), it was called a sous-double; a P4 equals one-third of the sum of its aliquot parts and was called a sous-triple; a P5 a sous-quadruple; etc. F. Marin Mersenne proposed to R. Descartes300 the problem to find a sous-double other than P3(1) = 120 = 233-5. The latter did not react on the question until seven years later. Mersenne301 mentioned (in the Epistre) the problem to find a P4, a P5 or a Pm, a P3 besides 120, and a rule to find as many as one pleases. He remarked (p. 211) that the P3 120, the P4 240 [for 30240?] and all other abundant numbers can signify the most fruitful natures. Pierre de Fermat302 referred in 1636 to his former [lost] letter in which he gave "the proposition concerning aliquot parts and the construction to find an infinitude of numbers of the same nature." He303 found the second P3, viz., P3(2)=672 = 253-7. Mersenne304 stated that Fermat found the 1 3 7 15... P3 672 and knew infallible rules and analysis 2 4 8 16... to find an infinitude of such numbers. He305 3 5 9 17... later gave [Fermat*s] method of finding such P3: Begin with the geometric »°°Oeuvres de Descartes, 1, Paris, 1897, p. 229, line 28, letter from Descartes to Mersenne, Oct or Nov., 1631. lolLes Preludes de VHannonie Universelle ou Questions Curiouses, Utiles aux Predicateurs, aux Theologiens, Astrologues, Medecins, & Philosophes, Paris, 1634. I020euvres de Fermat, 2, Paris, 1894, p. 20, No. 3, letter to Mersenne, June 24, 1636. J0z0euvres de Fermat, 2, p. 66 (French transl. 3, p. 288), 2, p. 72, letters to Mersenne and Roberval, Sept., 1636. IMHarmonie Universelle, Paris, 1636, Premiere Preface Generate (preceded by a preface of two pages), unnumbered page 9, remark 10. Extract in Oeuvres de Fermat, 2, 1894, 20-21. 108Mersenne, Seconde Partie de PHarmonie Universelle, Paris, 1637. Final subdivision: Nou- velles Observations Physiques et MathSmatiques, p. 26, Observation 13. Extract in Oeuvres de Fermat, 2, 1894, p. 21.