20 History of the Theory of Numbers. [Chap. I 
Johann Philipp Griison^^'' made the same criticism of Ozanam"^ and 
noted that, if 2"x is perfect and x is an odd prime, 
1+2+ . . .+2'' = 2'*x-a:-2x-. . .-2'*-^x = x. 
M. Fontana^"^ noted that the theorem that all perfect numbers are 
triangular is due to Maurolycus^^ and not to T. Maier (cf. Kraft^^). 
Thomas Taylor^"- stated that only eight perfect numbers have been 
found so far [the 8 listed are those of Mersenne^^j. 
J. Struve^^^ considered abundant numbers which are products ahc of 
three distinct primes in ascending order; thus 
ob+o+M-l 2 ^ .- 
— ; ; >C, >C + 1. 
ah-a-h-1 ' i_i_l_jL 
a h ab 
The case a^3 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 
^"^' >d+i. 
a6c — (a6+ac+6c+a+6+c+l)' 
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. Westerberg^*^ gave the factors of 2"='=1 for n = l,..., 32, and of 
10''±l,n = l,..., 15. 
O. Terquem^°^ Usted 2*^-1 and 2*^-1 as primes. 
L. WantzeP"® proved the remark of Kraft*^ that if A^i be the sum of the 
digits of a perfect number N>6 [of Euclid's type], and N2 the sum of the 
digits of A^i, etc., a certain iV, is unity. Since iV=l(mod 9), each Ni=l 
(mod 9), while the NiS decrease. 
V. A. Lebesgue^°^ 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 2"?/ V . . . , 
y'^" ■ ■ • +prrY = (1 +y+ ■ ■ ■ +y') d +^+ • • • +^') ■ • • » 
""Enthiillte Zaubereyen und Geheimnisse der Arithmetik, erster Theil, Berlin, 1796, p. 85, and 
Zusatz (end of Theil I). 
»<»Memorie dell' Istituto Nazionale Ital., mat., 2, pt. 1, 1808, 285-6. 
'°*The elements of a new arithmetical notation and of a new arithmetic of infinites, with an 
appendix .... of perfect, amicable and other numbers no less remarkable than novel, 
London, 1823, 131. 
^''Ueber die so gennannten numeri abundantes oder die Ueberfluss mit sich fiihrenden Zahlen, 
besonders im ersten Tausend unsrer Zahlen, Altona, 1827, 20 pp. 
*'**De factoribus numerorum compositorum dignoscendis, Disquisitio Acad. CaroUna, Lundae, 
1838. In the volume, Meditationum Math publice defendent C. J. D. Hill, Pt. II, 
1831. 
i^Nouv. Ann. Math., 3, 1844. 219 (cf. 553). 
^<*Ibid., p. 337. 
i"76id., 552-3. 
