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A positive integer is called perfect if it is equal to the sum of all its proper divisors, or equivalently if . If we denote by the sum of all positive divisors of , then is perfect if .
The notion of perfect number is very old and it is not known when perfect numbers were first studied. Its history does back to old Babylonian or Egyptian period. In that time, however, the definition used the notion of the “aliquot part” of a number . . Under an aliquot part of a number is a proper quotient of the number was understood. So for example the aliquot parts of are 1, 2 and 3, for , 2,and , but 6 is not an aliquot part of 6 since it is not a quotient different from the number itself (that is a proper quotient). A perfect number was defined to be one which is equal to the sum of its aliquot parts.
Perfect numbers were studied by Pythagoras and his followers, more for their mystical properties than for their number theoretic properties.
The first four perfect numbers are 6, 28, 496 and 8128. These were already known to early Greek mathematicians. One reason for the fascination of Greeks and their old Babylonian cultural predecessors could be facts that at that time there were known 6 “wanderers” circling the Earth (Mercury, Venus, Mars, Jupiter, and Saturn plus Moon) and 28 is the approximately the number of days it takes the Moon to make a complete orbit around the Earth.
Nicomachus of Gerasa (around 100 AD) gave in his famous Introductio Arithmetica a classification of numbers based on the concept of perfect numbers. He divided the numbers into three classes:
Nicomachus without proof to describes certain properties of perfect numbers:
The first known recorded mathematical on result perfect numbers can be found in Euclid's Elements (around 300 B.C). Proposition 36 of Book IX of the Elements gives an sufficient condition for construction of even perfect primes:
If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect.
In our notation: If is a prime, then the number is a perfect number.
The sufficiency of the proof known probably to Pythagoras is very simple: If is a prime, then all its aliquot parts are . Their sum is plus , that is .
A posthumous paper of L.Euler (1707-1783) contains the proof of the necessity, that is that every even perfect number is of the form given by Euclid.
Theorem (Euclid-Euler). If is an even perfect number, then it can be written in the form , where and are both primes. Conversely, if and are primes, then the number is perfect.
Note that a necessary condition for be a prime it that is also a prime. This result is often attributed to Fermat and Cataldi. Primes of the form are called Mersenne primes.
The proof of the Euclid part is easy, and be done by direct verification for the sum of divisors.
L.E.Dickson gave a very short proof of Euler’s necessary condition: If denotes the sum of all divisors of , then for a perfect number . Let with odd and . Then , by multiplicativity of 1 . Thus
(1) since , is an integer, say , is ,
(2) we also have
Since is the sum of divisors of , and , and this divisor does not appear in the sum, and . Moreover, since has only two divisors must be a prime. In other words, is a prime.
Euler’s original proof was similar.
Probably the first result on odd perfect numbers was a paper by Benjamin Pierce, Mathematical Instructor in Harward University, published in The New York Mathematical Diary, no.13, vol.2, pp.267-277 in 1832 that there can be no odd perfect number included in the form , , , where , and are prime numbers and greater than unity (cf. The American Mathematical Monthly, Vol. 28, No. 3. (Mar., 1921), p. 140.). In other words, an odd perfect number must have at least four distinct prime divisors.
1 | That is whenever are coprime integers. This fact was proved by Euler in his proof of converse of Euclid’s theorem. |
Cite this web-page as:
Štefan Porubský: Perfect Numbers.