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IV                                                    PREFACE,
noted in 1603 that 2P  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 228~1 has the factor 47, and 237-l the factor 223, while Euler observed in 1732 that 2291 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-1(2P-1) 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 267 1 is composite was proved by Lucas in 1876, while its actual factors were found by Cole hi 1903. The primality of 2611, a number of 19 digits, was established by Pervu&n in 1883, Seelhoff in 1886, and Hudelot in 1887. Both Powers and Fauquembergue proved hi 1911-14 that 289 1 and 21071 are primes. The primality of 231  1 and 21271 had been established by Euler and Lucas respectively. Thus 2P1 is known to be a prune, 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 2P~1 is known (pp. 15-31) to be composite for 32 prunes 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 ps2, where p is a prime. EulerJs proofs (p. 19) were published after his death. An 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 Mersenne, Fermat, St. Croix, Frenicle, and Descartes. Many new examples have 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 Thibit ben Korrah noted that Tht and 2ns are amicable numbers if ft = 3-2n-l, ^S-S""1 1 and s = 9'22n~*1~l are all primes, and n> 1. This result leads to amicable numbers for ft = 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.