Chap. I] PERFECT, MULTIPLY PERFECT, AND AMICABLE NUMBERS. 33 
G. Wertheim, Anfangsgriinde 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; Tenseignement 
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 multi- 
plicity m shall be designated by P^. Thus an ordinary perfect number is 
a P2. Although Robert Recorde^^ 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 KPs), 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. Descartes^°^ the problem to find a 
sous-double other than P^^^^ = 120 = 2^3-5. The latter did not react on the 
question until seven years later. 
Mersenne^"^ mentioned (in the Epistre) the problem to find a P4, a 
P5 or a P^, 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 Fermat^"- 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." He^^^ found the second 
P3, viz., P3<2) = 672 = 2^3-7. 
Mersenne^°* 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. He^°^ 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. 
^"iLes Preludes de I'Harmonie Universelle ou Questions Curiouses, Utiles aux Predicateurs, aux 
Theologiens, Astrologues, Medecins, & Philosophes, Paris, 1634. 
'•"Oeuvres de Fermat, 2, Paris, 1894, p. 20, No. 3, letter to Mersenne, June 24, 1636. 
"^Oeuvres de Fermat, 2, p. 66 (French transl. 3, p. 288), 2, p. 72, letters to Mersenne and 
Roberval, Sept., 1636. 
'"Harmonic Universelle, Paris, 1636, Premiere Preface Generale (preceded by a preface of two 
pages), imnumbered page 9, remark 10. Extract in Oeuvres de Fermat, 2, 1894, 20-21. 
"'Mersenne, Seconde Partie de I'Harmonie Universelle, Paris, 1637. Final subdivision: Nou- 
velles Observations Physiques et Math^matiques, p. 26, Observation 13. Extract in 
Oeuvres de Fermat, 2, 1894, p. 21. , 
