IV PREFACE. 
noted in 1603 that 2'' — 1 is composite if p is composite and verified that it is 
a prime for p = 13, 17, and 19; but he erred in stating that it is also a prime 
for p = 23, 29, and 37. In fact, Fermat noted in 1640 that 2'-'-l has the 
factor 47, and 2^'-l the factor 223, while Euler observed in 1732 that 
2'' — 1 has the factor 1103. Of historical importance is the statement made 
by Mersenne in 1644 that the first eleven perfect numbers are given by 
2P-i(2P_i) for p = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257; but he erred at 
least in including 67 and excluding 61, 89, and 107. That 2" — 1 is com- 
posite was proved by Lucas in 1876, while its actual factors were found by 
Cole in 1903. The primality of 2^^ — 1, a number of 19 digits, was estab- 
lished by Pervusin in 1883, Seelhoff in 1886, and Hudelot m 1887. Both 
Powers and Fauquembergue proved in 1911-14 that 2^^ — 1 and 2^°^ — 1 are 
primes. The primality of 2'^ — 1 and 2™ — 1 had been estabhshed by Euler 
and Lucas respectively. Thus 2^— 1 is known to be a prime, and hence lead 
to a perfect number, for the twelve values 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 
107 and 127 of p. Since 2^' — 1 is known (pp. 15-31) to be composite for 32 
primes p ^257, only the eleven values p = 137, 139, 149, 157, 167, 193, 199, 
227, 229, 241, 257 now remain in doubt. 
Descartes stated in 1638 that he could prove that every even perfect 
number is of Euclid's type and that every odd perfect number must be of the 
form ps^, where p is a prime. Euler's proofs (p. 19) were published after his 
death. Xd. immediate proof of the former fact was given by Dickson (p. 30). 
According to Sylvester (pp. 26-27), there exists no odd perfect number with 
fewer than six distinct prime factors, and none with fewer than eight if not 
divisible by 3. But the question of the existence of odd perfect numbers 
remains unanswered. 
A multiply perfect number, like 120 and 672, is one the sum of whose 
divisors equals a multiple of the number. They were actively investigated 
during the years 1631-1647 by IMersenne, Fermat, St. Croix, Frenicle, and 
Descartes. Many new examples hav^e been found recently by American 
writers. 
Two numbers are called amicable if each equals the sum of the aliquot 
divisors of the other, where an aliquot divisor of a number means a divisor 
other than the number itself. The pair 220 and 284 was known to the 
Pythagoreans. In the ninth century, the Arab Thabit ben Korrah noted 
that 2"/!« and 2"s are amicable numbers if /j=3-2''-l, t = 2>'2'^^-l and s = 
9.22"-! _i are all primes, and n> 1. This result leads to amicable numbers 
for n = 2 (giving the above pair), n = 4 and n = 7, but for no further value 
^ 200 of n. The chief investigation of amicable numbers is that by Euler 
who listed (pp. 45, 46) 62 pairs. At the age of 16, Paganini announced in 
1866 the remarkable new pair 1184 and 1210. A few new pairs of very 
large numbers have been found by Legendre, Seelhoff, and Dickson. 
