14 History of the Theory of Numbers. [Chap, i 
He stated that while perfect numbers end with 6 or 28, the proof by Bungus*' 
does not show that they end alternately with 6 and 28, since Bungus 
included imperfect as well as perfect numbers. The numbers 130816 and 
2096128, cited as perfect by Puteanus,^^ are abundant. After giving a 
table of the expanded form of 2" forn = 0, 1, . . . , 100, Broscius (p. 130, seq.) 
gave a table of the prime divisors of 2" — 1 (n = 1, . . . , 100), but showing no 
prime factor when n is any one of the primes, other than 11 and 23, less 
than 100. For n = ll, the factors are 23, 89; for n = 23, the factor 47 is 
given. Thus omitting unity, there remain only 23 numbers out of the first 
hundred which can possibly generate perfect numbers. Contrary to Car- 
dan, ^^ but in accord with Bungus,^^ there is (p. 135) no perfect number 
between 10* and 10\ Of Bungus' 24 numbers, only 10 are perfect (pp. 
135-140): those with 1, 2, 3, 4, 8, 10, 12, 18, 19, 22 digits, and given by 
2'-i(2'*-l) for n = 2, 3, 5, 7, 13, 17, 19, 29, 31, 37, respectively. The pri- 
maUty of the last three was taken on the authority of unnamed predecessors. 
There are only 21 abundant numbers between 10 and 100, and all of 
them are even; the only odd abundant number <1000 is 945, the sum of 
whose aliquot di\isors is 975 (p. 146). The statement by Lucas, Th^orie 
des nombres, 1, Paris, 1891, p. 380, Ex. 5, that 3^-5-79 [deficient] is the 
smallest abundant number is probably a misprint for 945 = 3^-5-7. This 
error is repeated in Encyclopedic Sc. Math., I, 3, Fas. 1, p. 56. 
Johann Jacob Heinlin^- (1588-1660) stated that the only perfect num- 
bers <4-10' are 6, 28, 496, 8128, 130816, 2096128, 33550336, and that all 
perfect numbers end alternately in 6 and 8. 
Andrea Tacquet^^ (Antwerp, 1612-1660) stated (p. 86) that Euclid's 
rule gives all perfect numbers. Referring to the 11 numbers given as 
perfect by Mersenne,^^ Tacquet said that the reason why not more have 
been found so far is the greatness of the numbers 2^ — 1 and the vast labor 
of testing their primaUty. 
Frenicle^ stated in 1657 that EucUd's formula gives all the even perfect 
numbers, and that the odd perfect numbers, if such exist, are of the form 
p/c^, where p is a prime of the form 4n+l [cf. Euler^^]. 
Frans van Schooten^^ (the younger, 1615-1660) proposed to Fermat 
that he prove or disprove the existence of perfect numbers not of Euclid's 
type. 
Joh. A. Leuneschlos^^ remarked that the infinite multitude of numbers 
contains only ten perfect numbers; he who will find ten others will know 
'*Joh. Jacobi Heinlini, Synopsis Math, praecipuas totius math .... Tubmgae, 1653. Synopsis 
Math. Universalis, ed. Ill, Tubingae, 1679, p. 6. English translation of last by Venterus 
Mandey, London, 1709, p. 5. 
"Arithmeticae Theoria et Praxis, Lovanii, 1656 and 1682 (same paging), [1664, 1704]. Hia 
opera math., Antwerpiae, 1669, does not contain the Arithmetic. 
"Correspondence of Chr. Huygens, No. 389; Oeuvres de Fermat, 3, Paris, 1896, p. 567. 
"Oeuvres de Huygens, II, Correspondence, No. 378, letter from Schooten to J. Wallis, Mar. 18, 
1658. Oeuvres de Fermat, 3, Paris, 1896, p. 558. 
"Mille de Quantitate Paradoxa Sive Admiranda, Heildelbergae, 1658, p. 11, XLVI, XLVII. 
