16 History of the Theory of Numbers. [Chap, i 
John Harris,"^ D. D., F. R. S., stated that there are but ten perfect 
numbers between unity and one million of millions. 
John Hiir^ stated that there are only nine perfect numbers up to a 
hundred thousand million. He gave (pp. 147-9) a table of values of 2" 
forn = l,. . ., 144. 
Christian Wolf" (1679-1754) discussed perfect numbers of the form 
y"x [where x, y are primes]. The sum of its aliquot parts is 
l+y+ . . . +i/"+a:+?/x+ . . . +i/""'x, 
which must equal y'^x. Thus 
x = {l-\-y+ . . . +2/")M d = y^-l-y- . . . -7/""^ 
He stated* that x is an integer only when d = l, and that this requires 
y = 2, x = l +2+ . . . +2". Then if this x is a prime, 2"x is a perfect number. 
This is said to be the case forn = 8 and n = 10, since 2^ — 1 = 51 1 and 2^' — 1 = 
2047 are primes, errors pointed out bj^ Euler.^^ A. G. Kastner^^ was not 
satisfied with the argument leading to the conclusion y = 2. 
Jacques Ozanam^^ listed as perfect numbers 
2(4-1), 4(8-1), 16(32-1), 64(128-1), 256(512-1), 1024(2048-1),. . . 
without expUcit mention of the condition that the final factor shall be prime, 
and stated that perfect numbers are rare, only ten being known, and all 
end in 6 and 8 alternately. [Criticisms by Montucla,®^ Gruson.-^°°] 
Johann Georg Liebnecht^° said there were scarcely 5 or 6 perfect num- 
bers up to 4.10"; they always end alternately in 6 and 8. 
Alexander ]Malcolm^^ observed that it is not yet proved that there is no 
perfect number not in Euclid's set. He stated that, if pA is a perfect 
number, where p is a prime, and if M<p and M is not a factor of A, then 
MM is an abundant number [probably a misprint for MA, as the condi- 
tions are satisfied when p = 7, .4=4, M = 5, and MA =20 is abundant, while 
Af^ = 25 is deficient]. 
Christian Wolf^- made the same error as Casper Ens.^^ 
^'Lexicon Technicum, or an Universal English Dictionarj' of Arts and Sciences, vol. I, London, 
1704; ed. 5, vol. 2, London, 1736. 
"Arithmetik, London, ed. 2, 1716, p. 3. 
^'Elementa Matheseos Universae, Halae Magdeburgicae, vol. I, 1730 and 1742, pp. 383-^, of 
the five volume editions [first printed 1713-41]; vol. I, 1717, 315-6, of the two volume 
edition. Quoted, with other errors, Ladies' Diary, 1733, Q. 166; Leybourn's ed., 1, 
1817, 218; Button's ed., 2, 1775, 10; Diarian Repository, by Soc. Math., 1774, 289. 
*"Jam ut X sit numerus integer, nee in casu speciali, si y per numerum explicetur, numerus 
partium aliquotarum diversus sit a numero earundem in formula general!; necesse est ut 
d = l." 
"Math. Anfangsgrlinde, I, 2 (Fortsetzung der Rechnenkunst, ed. 2, 1801, 546-8). 
"Recreations math., new ed. of 4 vols., 1723, 1724, 1735, etc., I, 29-30. 
'"Grund-Satze der gesammten Math. Wiss. u. Lehren, Giessen u. Franckfurt, 1724, p. 21. 
*iA new system of arithmetik, theoretical & practical, London, 1730, p. 394. 
"Mathematisches Lexicon, I, 1734 (under Vollkommen Zahl). 
