Chap. I] PERFECT, MULTIPLY PERFECT, AND AmICABLE NUMBERS. 15 
that he has surpassed all analysis up to the present. Goldbach" called 
Euler's attention to these remarks and stated that they were probably 
taken from Mersenne, the true sense not being followed. 
Wm. Leybourn^^ hsted as the first ten perfect numbers and the twentieth 
those which occur in the table of Bungus.^^ "The number 6 hath an emi- 
nent Property, for his parts are equal to himself." 
Samuel Tennulius, in his notes (pp. 130-1) on lamblicus,^ 1668, stated 
that the perfect numbers end alternately in 6 and 8, and included 130816 = 
256-511 and 2096128 = 1024-2047 among the perfect numbers. 
Tassius®^ stated that all perfect numbers end in 6 or 8. Any multiple 
of a perfect or abundant number is abundant, any divisor of a perfect 
number is deficient. He gave as the first eight known perfect numbers the 
first eight listed by Mersenne.^" 
Joh. Wilh. Pauli^° (Philiatrus) noted that if 2" — 1 is a prime, n is, but 
not conversely. For n = 2, 3, 5, 7, 13, 17, 19, 2"-l is a prime; but 2^^-l 
is divisible by 23, 2^^ — 1 by 47, and 2*^ — 1 by 83, the three divisors being 
2n+l. 
G. W. Leibniz'^^ quoted in 1679 the facts stated by Pauli and set himself 
the problem to find the basis of these facts. Returning about five years 
later to the subject of perfect numbers, Leibniz implied incorrectly that 
2^^ — 1 is a prime if and only if p is. 
Jean Prestet^^ (tl690) stated that the fifth, . . . , ninth perfect numbers are 
23550336 [for 33. . .], 8589869056, 137438691328, 238584300813952128 [for 
2305. . .39952128], 2''^-2^'\ 
[Hence 2'*-^(2'*-l) for 7i = 13, 17, 19, 31, 257. The numerical errors were 
noted by E. Lucas,i24 p 7g4 j 
Jacques Ozanam^^ (1640-1717) stated that there is an infinitude of perfect 
numbers and that all are given by Euclid's rule, which is to be applied only 
when the odd factor is a prime. 
Charles de Neuveglise^^ proved that the products 3-4, . . ., 8-9 of two 
consecutive numbers are abundant. All multiples of 6 or an abundant 
number are abundant. 
"Correspondence Math. Phy8.,ed.,Fus8, 1, 1843; letters to Euler, Oct. 7, 1752 (p. 584), Nov. 18 
(p. 593). 
'^Arithmetical Recreations; or Enchriridion of Arithmetical Questions both Delightful and 
Profitable, London, 1667, p. 143. 
"Arithmeticae Empiricae Compendium, Johannis Adolfi Tassii. Ex recensione Henrici Siveri, 
Hamburgi, 1673, pp. 13, 14. 
^"De nvmiero perfecto, Leipzig, 1678, Magister-disputation. 
"Manuscript in the Hannover Library. Cf. D. Mahnke, Bibhotheca Math., (3), 13, 1912-3, 
53-4, 260. 
"Nouveaux elemens des Mathematiques, ou Principes generaux de toutes les sciences, Paris, 
1689, I, 154-5. 
"Recreations mathematiques et physiques, Paris and Amsterdam, 2 vols., 1696, I, 14, 15. 
"Traits methodique et abreg6 de toutes les mathematiques, Trevoux, 1700, tome 2 (L'arith- 
m^tique ou Science des nombres), 241-8. 
