18 History of the Theory of Numbers. [Chap, i 
Johann Christoph Heilbronner^^ stated that the perfect numbers up to 
4-10' are 6, 28, 496, 8128, 130816,2096128. "The fathers of the early church 
and many wTiters always held this number 6 in high esteem. God com- 
pleted the creation in 6 days and since all things created by Him came out 
perfect, he wished the work of creation completed according to the number 6 
as being a perfect number." 
L. Euler" deduced from Fermat's theorem, which he here proved by 
use of the binomial theorem, the result* that, if m is a prime, 2"* — 1, when 
composite, has no prime factors other than those of the form wn+l. 
J. Landen^^ noted that 196 is the least number 4a;'*, where x is prime, the 
sum of whose ahquot parts exceeds the number by 7. 
L. Euler^^ gave a table of the prime factors of 2" — 1 for n^37. 
C. N. de Winsheim^° noted that 2^'^ — 1 has the factor 2351, and stated 
that 2" — 1 is a prime for n = 2, 3, 5, 7, 13, 17, 19, 31, composite for the 
remaining n<48, but was doubtful as to n = 41, thus reducing the Hst of 
perfect numbers given by Euler^ by one or perhaps two. He suspected 
that n = 41 leads to an imperfect number since it was excluded by the acute 
Mersenne,^° who gave instead 2^^(2^" — 1) as the ninth perfect number. He 
remarked that the basis of Mersenne's assertion is doubtless to be found in 
the stupendous genius of Mersenne which perhaps recognized more truths 
than he could demonstrate. He discussed the error of Hansch^ that 2" — ! 
is a prime if n is a prime ^ 79. 
G. W. Kraft^^ considered perfect numbers AP, where P is a prime [not 
dividing A]. Thus a{P-\-l)=2AP, where a is the sum of all the divisors 
of A. Hence a/ {2 A— a) equals the prime P. Let 2A— a = l, a property 
holding for A =2"". Then P = 2"'+^ — 1 and the resulting numbers are of 
Euchd's type. 
L. Euler,^- in a letter to Goldbach, October 28, 1752, stated that he 
knew only seven perfect numbers, viz., 2p~^(2^ — 1) for p = 2, 3, 5, 7, 13, 17, 
19, and was uncertain whether 2^^ — 1 is prime or not (a factor is necessarily 
of the form 64n+l and none are <2000). 
^^Historia matheseos universae. Accedit recensio elementorum compendiorum et openim math, 
atque historia arithmetices ad nostra tempora, Lipsiae, 1742, 755-6. There is a 63-page 
Ust of arithmetics of the 16th century. 
«^^ovi Comm. Ac. Petrop., 1, 1747-8, 20; Comm. .\rith., I, 56, §39. 
*We may simpUfy the proof by using the fact that 2 belongs to an e.xponent e modulo p (p a 
prime) such that e divides p — 1. For, if p is a factor of 2'"— 1, m is a multiple of e, whence 
e equals the prime m. Thus p — 1 =n7«. If we take m>2, we see that n is even since 
p is odd and conclude with Fermat^' that, if m is an odd prime, 2"»— 1 is divisible by no 
primes other than those of the form 2km + l. 
"•Ladies' Diary, 1748, Question 305. The Diarian Repository, Collection of all the mathe- 
matical questions from the Ladies' Diary, 1704-1760, by a society of mathematicians, 
London, 1774, 509. Button's The Diarian Miscellany (from Ladies' Diarj-, 1704-1773), 
London, 1775, vol. 2, 271. Leyboiu-n's Math. Quest, proposed in Ladies' D., 2, 1817, 
9-10. 
"Opuscula varii argumenti, Berlin, 2, 1750, 25; Comm. Arith., 1, 1849, 104. 
•"Novi Comm. Ac. Petrop., 2, 1751, ad annum 1749, mem., 68-99. 
"/bid., mem., 112-3. 
•^Corresp. Math. Phys. (ed., Fuss), I, 1843, 590, 597-8. 
