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Perfect numbers

A positive integer typeset structure is called perfect if it is equal to the sum of all its proper divisors, or equivalently if typeset structure. If we denote by typeset structure the sum of all positive divisors of typeset structure , then typeset structure is perfect if typeset structure.

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  typeset structure are 1, 2 and 3, for typeset structure, 2typeset structure,and typeset structure, 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  typeset structure is a prime, then the number typeset structure is a perfect number.

The sufficiency of the proof known probably to Pythagoras is very simple: If typeset structureis a prime, then all its aliquot parts are typeset structure. Their sum is typeset structureplus typeset structure, that is typeset structure.  

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  typeset structure is an even perfect number, then it can be written in the form typeset structure, where typeset structure and typeset structure are both primes. Conversely, if typeset structure and typeset structure are primes, then the number typeset structure is perfect.

Note that a necessary condition for typeset structure be a prime it that typeset structure is also a prime. This result is often attributed to Fermat and Cataldi. Primes of the form typeset structure 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 typeset structure denotes the sum of all divisors of typeset structure, then typeset structure for a perfect number typeset structure. Let typeset structure with odd typeset structure and typeset structure. Then typeset structure, by multiplicativity of typeset structure 1 .  Thus

(1) since typeset structure, typeset structure is an integer, say typeset structure, is  typeset structure,

(2) we also have

σ(q) = (2^(n + 1) q)/(2^(n + 1) - 1) = (2^(n + 1) - 1)/(2^(n + 1) - 1) q + q/(2^(n + 1) - 1) = q + d .

Since typeset structure is the sum of divisors of typeset structure, and typeset structure, and this divisor does not appear in the sum, typeset structure and  typeset structure. Moreover, since typeset structure has only two divisors typeset structure must be a prime. In other words, typeset structure 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 typeset structure, typeset structure, typeset structure, where typeset structure, and typeset structure 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.

Notes

1 That is typeset structure whenever typeset structure 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.

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