20 HISTORY OF THE THEORY OF NUMBERS. [CHAP, i Johann Philipp Griison100 made the same criticism of Ozanam79 and noted that, if 2nx is perfect and x is an odd prime, M. Fontana101 noted that the theorem that all perfect numbers are triangular is due to Maurolycus41 and not to T. Maier (cf. Kraft85). Thomas Taylor102 stated that only eight perfect numbers have been found so far [the 8 listed are those of Mersenne60]. J. Struve103 considered abundant numbers which are products dbc of three distinct primes in ascending order; thus a b ab The case afe3 is easily excluded, also a =2, 6^5 [except 2-5-7]. For a = 2, 6=3, c any prime >3, 6c is abundant. Next, abed is abundant if 2abc "*" ' For a =2, 6 = 3, c=5 or 7, and for a =2, 6 = 5, c = 7, abed is abundant for any prime d [>c]. Of the numbers ^ 1000, 52 are abundant. J. Westerberg104 gave the factors of 2n=*=l for n = l,..., 32, and of 10n=tl, n = l,..., 15. 0. Terquem105 listed 241-1 and 247-l as primes. L. Wantzel106 proved the remark of Kraft88 that if NI be the sum of the digits of a perfect number ]V>6 [of Euclid's type], and N2 the sum of the digits of NI, etc., a certain Nt is unity. Since AT=l(mod 9), each N^l (mod 9), while the Njs decrease. V. A. Lebesgue107 stated that he had a proof that there is no odd perfect number with fewer than four distinct prime factors. For an even perfect number 2yV. . ., l°°Enthiillte Zaubereyen und Geheimnisse der Arithmetik, erster Theil, Berlin, 1796, p. 85, and Zusatz (end of Theil I). 101Memorie dell' Istituto Nazionale Ital., mat., 2, pt. 1, 1808, 285-6. 1MThe elements of a new arithmetical notation and of a new arithmetic of infinites, with an appendix ---- of perfect, amicable and other numbers no lees remarkable than novel, London, 1823, 131. 103Ueber die so gennannten numeri abundantes oder die Ueberfluss mit eich fiihrenden Zahlen, besonders im ersten Tausend unsrer Zahlen, Altona, 1827, 20 pp. 104De factoribus numerorum compositorum dignoscendis, Disquisitio Acad. Carolina, Lundae, 1838. In the volume, Meditationum Math ..... publice defendent C. J. D. Hill, Pt. II, 1831. 1MNouv. Ann. Math., 3, 1844, 219 (cf. 553). lw/6td., p. 337. ™Ibid., 552-3.