16 HISTOHY OF THE THEORY OP NUMBERS, [CHAP. I John Harris,75 D. D., F. R. S., stated that there are but ten perfect numbers between unity and one million of millions. John Hill76 stated that there are only nine perfect numbers up to a hundred thousand million. He gave (pp. 147-9) a table of values of 2tt for n = l, . . ., 144. Christian Wolf77 (1679-1754) discussed perfect numbers of the form ynx [where x, y are primes]. The sum of its aliquot parts is • . . +yn+x+yx+ . , . +yn~lx, which must equal ynx. Thus He stated* that x is an integer only when d=l, and that this requires y = 2, £ = 1 +2 + - . . +2n. Then if this x is 'a prime, 2nx is a perfect number. This is said to be the case f or n = 8 and n = 10, since 29 — 1 = 511 and 211 — 1 = 2047 are primes, errors pointed out by Euler.83 A. G. Kastner78 was not satisfied with the argument leading to the conclusion y— 2. Jacques Ozanam79 listed as perfect numbers 2(4-1), 4(8-1), 16(32-1), 64(128-1), 256(512-1), 1024(2048-1),. . . without explicit 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," Griison.100] Johann Georg Liebnecht80 said there were scarcely 5 or 6 perfect numbers up to 4.107; they always end alternately in 6 and 8. Alexander Malcolm81 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 conditions are satisfied when p - 7, A =4, M =5, and MA ==20 is abundant, while M2=* 25 is deficient]. Christian Wolf82 made the same error as Casper Ens.51 "Lexicon Technicum, or an Universal English Dictionary 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. 383-4, 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. *"Jain ut x sit numerus integer, nee in casu speciali, si y per numerum explicetur, numerus partium aliquotarum diversus sit a numero earundem in formula generali; necesse est ut d = l." "Math. Anfangsgrtinde, I, 2 (Fortsetzung der Rechnenkunst, ed. 2, 1801, 546-8). "Recreations math., new ed. of 4 vols., 1723, 1724, 1735, etc., I, 29-30. 80Grund-Satze der gesammten Math. Wiss. u. Lehren, Giessen u. Franckfurt, 1724, p. 21. 81A new system of arithmetik, theoretical & practical, London, 1730, p. 394. "Mathematisches Lexicon, I, 1734 (under Vollkommen Zahl).