Perfect number

In Mathematical and more precisely in Arithmetic modular, a perfect number is an integer N strictly higher than 1 which is equal to the strict sum of its Diviseur S, in other words, such as $\ sigma (N) = 2n \,$ where $\ sigma (N) \,$ is the sum of the positive whole dividers of N , N included/understood.

Properties

The mathematician Euclide, at third century BC, discovered and proved that if $M=2^p-1 \,$ is first, then $M \ cdot \ left (\ frac {M+1} {2} \ right) = 2^ {p-1} (2^p - 1)$ is perfect.

In addition, Leonhard Euler, at the 18th century, proved that any even perfect number is form suggested by Euclide. The search for even perfect numbers is thus related to that of the prime numbers of Mersenne (prime numbers of the form 2^p-1).

It is established that any even perfect number ends in one 6 or one 8, but not inevitably in alternation.

In 2000, Douglas Iannucci showed in Journal off Integer Sequences 3,2000, Article 00.1.2 that all the perfect even numbers are numbers of Kaprekar bases two of them.

Examples

The first 4 perfect numbers are known since antiquity. Since, the total passed to 44 perfect numbers only (at the 9/11/2006).

The first twelve perfect numbers are:

6 = 1 + 2 + 3
28 = 1 + 2 + 4 + 7 + 14
496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8.128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1.016 + 2.032 + 4.064
33.550.336
8.589.869 056
137.438.691 328
2.305.843 008.139.952 128
2.658.455 991.569.831 744.654.692 615.953.842 176
191.561.942 608.236.107 294.793.378 084.303.638 130.997.321 548.169.216
13.164.036 458.569.648 337.239.753 460.458.722 910.223.472 318.386.943 117.783.728 128
14.474.011 154.664.524 427.946.373 126.085.988 481.573.677 491.474.835 889.066.354 349.131.199 152.128

The complete listing is on the site of J. Pedersen.

Curiosity

By dividing each equality above by the perfect number corresponding, one discovers a property of some Egyptian fractions:

1 = 1/6 + 1/3 + 1/2
1 = 1/28 + 1/14 + 1/7 + 1/4 + 1/2
and so on

Conjectures

The assertion " there does not exist any perfect number impair" is a Conjecture. Indeed, one is unaware of if there exist odd perfect numbers; such a number must have at least 11 factors first distinct of which at least is higher than 300.000; an odd perfect number must be higher than $10^ {300}$ . The hope to find one day a perfect number odd is not completely excluded, in the direction where there exist odd numbers almost perfect (spoof perfect numbers). Let us consider for example the following number:

198585576189 = 22021 × 3² × 7² × 11² × 13²

Such a number would be perfect if the 22021 were a factor first, which is not the case.

It is not known if there exists an infinity of prime numbers of Mersenne. It is not known either if there exists an infinity of perfect numbers.

See too

abundant Number
friendly Numbers
defective Number
almost perfect Number
sociable Prime number
Number

External bonds

Geometry of the prime numbers and the perfect numbers

Simple: Perfect number

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