THE

PERFECT AND AMICABLE NUMBERS

By Professor L. E. DICKSON

UNIVERSITY OF CHICAGO

HE two types of numbers in our title have had a continuous history extending from the early Greeks to date, and may be justly called the most human of all numbers. To them were early attributed certain social qualities, and later also ethical import, while mystics of the middle ages believed that they possessed special powers as talismans. Continuously for twenty centuries a wide-spread interest has been taken in the purely numerical questions and puzzle problems which arose in the study of these remarkable numbers. We shall present here the more essential facts and fancies in the quaint history of these most human of all numbers.

A perfect number is one which equals the sum of its aliquot divisors, i. e., divisors which are less than the number itself. Thus 6 is a perfect number, being equal to the sum of 1, 2 and 3. The Pythagoreans called the perfect number 6 marriage "on account of the integrity of its parts and the agreement existing in it." In his De Nuptiis Philologiae, Martiano Capella stated that the perfect number 6 is attributed to Venus; for, it is made by the union of the two sexes, that is from triad, which is male since it is odd, and from diad, which is feminine since it is even.

To understand a later allusion, we need the definitions given by Nicomachus (about 100 A. D.) that 12 is an abundant number since the sum 16 of its aliquot divisors exceeds 12, while 8 is a deficient number since the sum 7 of its aliquot divisors is less than 8. He remarked that perfect numbers are between excess and deficiency, as consonant sound between acuter and graver sounds. In his De Civitate Dei, Aurelius Augustinus (354-430 A. D.) remarked that, 6 being the first perfect number, God effected the creation in 6 days rather than at once, since the perfection of the work is signified by the number 6. Alcuin (735-804), of York and Tours, added the remark that the second origin of the human race arose from the deficient number 8; indeed, in Noah's ark there were 8 souls from which sprung the entire human race, showing that the second origin was more imperfect than the first, which was made according to the perfect number 6. Rabbi Josef b. Jehuda Ankin, at the end of the twelfth century, recommended the study of perfect numbers in the program of education laid out in his book Healing of Souls. We shall return presently to the history of the arithmetical study of perfect numbers.

Two numbers, like 220 and 284, are called amicable if each equals the sum of the aliquot divisors of the other. Iamblichus states that Pythagoras remarked that the aliquot parts [divisors] of each of the numbers 220 and 284 have the power to generate the other, according to the rule of friendship; and that, when asked what is a friend, he replied "another I," which is shown in these numbers. According to Rau Nachshon (ninth century A. D.), Jacob prepared wisely his present to Esau of 200 she-goats and 20 he-goats, 200 ewes and 20 rams (Genesis, xxxii, 14), since this number 220 of goats is a hidden secret, being one of a pair of numbers such that the parts of it are equal to the other one 284, and conversely; and Jacob had this in mind; this has been tried by the ancients in securing the love of kings and dignatories.

Ibn Khaldoun related that "persons who have concerned themselves with talismans affirm that the amicable numbers 220 and 284 have an influence to establish a union or close friendship between two individuals. To this end a theme is prepared for each individual, one during the ascendency of Venus, when that planet is in its exaltation and presents to the moon an aspect of love or benevolence; for the second theme, the ascendency should be in the seventh. On each of the themes is written one of the specified numbers, the greater being attributed to the person whose friendship is sought." The Arab el-Magriti of Madrid (who died in 1007) related that he had put to the test the erotic effect of "giving anyone the smaller number 220 to eat, and himself eating the larger number 284."

The writer verified a few years ago that the only pairs of amicable numbers in which the smaller number does not exceed 6232 are 220 and 284, 1184 and 1210, 2620 and 2924, 5020 and 5564, 6232 and 6368. The second pair was discovered in 1866 by N. Paganini at the age of 16. It was missed by Euler, who in 1750 made the chief investigation of amicable numbers and listed 62 pairs. It would be easy, but not very interesting, to obtain further pairs, by the methods explained by Euler, by employing a table of prime numbers extending beyond the limit 100,000 of the table accessible to him. A prime number is one, like 2, 3, 5, 7, 11, which has no divisor other than itself and unity.

Let us return to perfect numbers and consider the main facts in their arithmetical history. We shall write 23 for 8 and 2p for the product of p factors 2.

In his famous Greek text on geometry, Euclid proved that 2p-1 (2-1) is a perfect number if 2-1 is a prime. For p=2, 3, 5, 7, the values of 2-1 are 3, 7, 31, 127, which are all primes, so that 6, 28, 496 and 8126 are perfect numbers. These four numbers were mentioned explicitly by Nicomachus, who noted that they are the only perfect numbers in the respective intervals between 1, 10, 100, 1000

and 10,000. The last fact was twisted by Iamblichus (about 283-330 A. D.) and many later writers into the erroneous conclusion that there exists one and only one perfect number between any two successive powers of 10.

The fifth perfect number 33,550,336 was first given in 1456 in the manuscript Codex lat. Monacensis 14908; it corresponds to the value 13 of p in Euclid's formula. Many writers listed false perfect numbers, due to their belief that 2P-1 is a prime for every value of p which is odd (i. e., not even). But Regius, in his Arithmetic printed at Strasbourg in 1536, noted that 29-1-511-7:73 and 211-1= 2047-23.89 are not primes, while 213—1—8191 is a prime and leads to the above fifth perfect number.

Cataldi, who founded at Bologna the most ancient known academy of mathematics, verified in 1603 that 21-1=131,071 and 21o-1 are primes by the unnecessarily laborious work of trying as possible divisors each prime less than their respective square roots. He therefore concluded correctly that the sixth and seventh perfect numbers are 216 (217-1)=8,589,869,056 and 218 (219-1). But he stated erroneously that 2-1 is a prime when p=23, 29 and 37. In fact, Fermat noted in 1640 that 223-1 has the factor 47, and 237-1 the factor 223, while Euler observed in 1732 that 229-1 has the factor 1103. These errors cast doubt on the validity of Cataldi's claim that 231-1 is a prime.

Fermat, who was a member of the parliament of Toulouse and an arithmetician of the highest ability, stated in 1640 the important fact that if p is a prime, 2o-1 is divisible by no primes other than those of the form 2 k p+ 1. Hence if 231-1 were not a prime each of its prime factors would be of the form of 62k+1, so that it is unnecessary to consider most of the trial divisors tested by Cataldi. We shall see below that Euler knew a general principle which eliminates half of the trial divisors required by Fermat's rule. Closely related to all these facts is the second proposition stated by Fermat that, if q is an odd prime, 2-1 is divisible by q, and his generalization that if q is any prime and n is any whole number not divisible by q, then n9-1-1 is divisible by q. This result, which is the basis of the modern theory of numbers, is known as Fermat's theorem.

Mersenne, who acted as intermediary in the extensive correspondence between Fermat, Frenicle, Descartes, and other expert arithmeticians, quoted various arithmetical results due to them in his curious books, Harmonie Universelle, Cogitata Physico Mathematica, etc., published in 1634-1647. But when he made the oft-quoted statement that the first eleven perfect numbers are given by 2P-1 (2-1) for p=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257, he was relying only in part upon more modest facts communicated to him by his able correspond

ents, but mainly upon some unlucky personal guessing as to the hidden mystery of prime numbers. He pretended to know that the number 2127-1 of 39 figures is a prime, that the number 2257-1 (which exceeds the square of the preceding vast number) is a prime, and that all of the intermediate numbers 2-1 are composite,—and yet he admitted that "to tell if a given number of 15 or 20 figures is prime or not, all time would not suffice for the test, whatever use is made of what is already known." More than two centuries later it was shown that Mersenne erred at least in including p=67 in his list and in excluding the values 61, 89 and 107 of p. The fact that 267-1 is composite was proved by Lucas in 1876, while its actual factors were found by Cole in 1903. The fact that 261-1, a number of 19 figures, is a prime was proved independently by Pervusin in 1883, Seelhoff in 1886, Hudelot in 1887, and Cole in 1903. Both Powers and Fauquembergue proved in 1911-1914 that 289-1 and 21o7—1 are primes. It is not surprising that Mersenne's guesses were erroneous, but it is quite surprising that his errors have been detected, thanks to the powerful modern methods of testing whether or not 2-1 is a prime.

Euler, one of the greatest mathematicians of all ages, proved in 1732 that, if p=4n-1 and 8n-1 are primes, 2P-1 has the factor 8n-1, so that 2P-1 is not a prime when p=11, 23, 83, 131, 179, 191, 239, 251, etc. He noted that 248-1 has the factor 431 and that 27—1 has the factor 439. In 1741 he found that 2-1 has the factor 2351, a case which had earlier earlier deceived him. In 1772 he proved that 231-1 is a prime. According to Fermat's result, every prime factor p is of the form 62k+1. But p divides 2 (231—1)=a2—2, where a=216, and Euler knew that every prime which divides any number of this form a2-2 is of one of the forms 8n+1, 8n+7. Hence in p=62k+1, k must be of the form 4m or 4m+1. Euler therefore considered only the possible prime factors of the form 248m+1 or 248m+63.

11119

In 1877 Lucas proved that A=2127-1 is a prime by a powerful new method. For so great a number (of 39 figures), it is clearly impracticable to test directly all the primes 2:127 k+1 as possible factors, or even the half of them under Euler's simplification of Fermat's method. Instead, we use the recurring series 1, 1, 2, 3, 5, 8, 13, which Leonardo Pisano had first employed in 1202 to find the number of offspring of a pair of rabbits. Each term of this series equals the sum of the preceding two terms. Write u for the kth term. One of Lucas's tests for primality applies to numbers having the remainder 3 or 7 when divided by 10. Since 25 32 has the same remainder as 2, we may suppress multiples of 4 in the exponent of a power of 2 when finding the remainder on division by 10. Hence the remainder from our A is 23—1=7. Writing k=2", Lucas verified that 127 is the

1

k

least positive whole number n such that u is divisible by A; hence A is a prime by his test. Similar tests for primality due to Lucas were used by Fauquembergue and Powers independently in 1914-1917 to prove that 289-1 and 2107-1 are primes and that 2-1 is not prime when p=101, 103, 109, the actual factors not being found.

The further known results are exhibited in the following table which gives the least prime factor of 2-1 for the various primes p, and the name of the discoverer of the factor and the date:

To summarize the results quoted above, 2P-1 is known to be composite for thirty-two of the primes p<257, but to be a prime and hence to lead to a perfect number 2P-1 (2-1) for the following twelve values: p=2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127. Hence only the following eleven values now remain in doubt: p=137, 139, 149, 157, 167, 193, 199, 227, 229, 241, 257.

All that precedes relates to perfect numbers which are given by Euclid's formula 2P-1 (2-1). Is every perfect number of this form? In 1638 Descartes stated that he could prove that every even perfect number is given by Euclid's formula. By far the simplest proof is that published by the writer in 1910. Let 2oq be an even perfect number, where q is odd. All the divisors of 2oq are obtained by multiplying each of the divisors 2o, 2o-1, -, 2, 1 of 2a by each of the divisors of q. Hence the sum of the divisors of 2oq is ms, where m=2a+--+2+1 and s is the sum of the divisors of q. If we multiply m by 2-1, we see that all but two terms of the product cancel, so that m=2n+1-1. By definition, a perfect number equals the sum of all its divisors other than itself, whence

2"q=ms-2"q,

2a+1q=ms (2+11) s Dividing by 2+1-1, we get s=q+d, where d=q/ (20+1-1). Thus the whole number d=s-q is a divisor of q. Since the sum s of all the divisors of q reduces to q+d, q and d are the only divisors of q, so that d=1 and q is a prime. From d=1, we get q=2n+1-1. Hence every even perfect number 2"q is of Euclid's type.

Whether or not there exists an odd perfect number has never been decided. If one exists, it must be of the form ps2, where p is a prime,

VOL. XII.-23.