What is Perfect number?

In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.

Today the usual definition of a perfect number is in terms of its divisors, but early definitions were in terms of the 'aliquot parts' of a number.

The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum.

An aliquot part of a number is a proper quotient of the number. 
So for example the aliquot parts of 10 are 1, 2 and 5. These occur since 1 = ¹⁰/₁₀, 2 = ¹⁰/₅, and 5 = ¹⁰/₂.
 Note that 10 is not an aliquot part of 10 since it is not a proper quotient, i.e. a quotient different from the number itself. 
A perfect number is defined to be one which is equal to the sum of its aliquot parts.

The first few perfect numbers are 6, 28, 496 & 8128.

Euclid proved that 2^p−1(2^p − 1) is an even perfect number whenever 2^p − 1 is prime.

For example, the first four perfect numbers are generated by the formula 2^p−1(2^p − 1), with p a prime number, as follows:

for p = 2:   2¹(2² − 1) = 2 × 3 = 6

for p = 3:   2²(2³ − 1) = 4 × 7 = 28

for p = 5:   2⁴(2⁵ − 1) = 16 × 31 = 496

for p = 7:   2⁶(2⁷ − 1) = 64 × 127 = 8128.

Prime numbers of the form 2^p − 1 are known as Mersenne primes.

For 2^p − 1 to be prime, it is necessary that p itself be prime. 
However, not all numbers of the form 2^p − 1 with a prime p are prime; for example, 2¹¹ − 1 = 2047 = 23 × 89 is not a prime number. 

In fact, Mersenne primes are very rare—of the 2,610,944 prime numbers p up to 43,112,609, 2^p − 1 is prime for only 47 of them.

There is a one-to-one correspondence between even perfect numbers and Mersenne primes; each Mersenne prime generates one even perfect number, and vice versa. This result is often referred to as the Euclid–Euler theorem.

 It is not known whether there are infinitely many perfect numbers, nor whether there are infinitely many Mersenne primes.

Every even perfect number is a pernicious number.

Every even perfect number is also a practical number .

It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. 

All even perfect numbers have a very precise form; odd perfect numbers either do not exist or are rare.

Some results :

• The only even perfect number of the form x³ + 1 is 28 .
• 28 is also the only even perfect number that is a sum of two positive cubes of integers.
• The reciprocals of the divisors of a perfect number N must add up to 2 (to get this, take the definition of a perfect number, ${\displaystyle \sigma _{1}(n)=2n}$, and divide both sides by n):
  • For 6, we have ${\displaystyle 1/6+1/3+1/2+1/1=2}$;
  • For 28, we have ${\displaystyle 1/28+1/14+1/7+1/4+1/2+1/1=2}$, etc.
• The number of divisors of a perfect number (whether even or odd) must be even, because N cannot be a perfect square.
  • From these two results it follows that every perfect number is an Ore's harmonic number.
• The even perfect numbers are not trapezoidal numbers; that is, they cannot be represented as the difference of two positive non-consecutive triangular numbers. There are only three types of non-trapezoidal numbers: even perfect numbers, powers of two, and the numbers of the form ${\displaystyle 2^{n-1}(2^{n}+1)}$ formed as the product of a Fermat prime ${\displaystyle 2^{n}+1}$ with a power of two in a similar way to the construction of even perfect numbers from Mersenne primes.
• The number of perfect numbers less than n is less than ${\displaystyle c{\sqrt {n}}}$, where c > 0 is a constant. In fact it is ${\displaystyle o({\sqrt {n}})}$, using little-o notation.
• Every even perfect number ends in 6 or 28, base ten; and, with the only exception of 6, ends in 1, base 9. Therefore in particular the digital root of every even perfect number other than 6 is 1.
• The only square-free perfect number is 6.

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