Question

Who gave the concept of perfect numbers , and also how can we find them

Answer 1

In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. σ1(n) = 2n.

This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called τέλειος ἀριθμός (perfect, ideal, or complete number). Euclid also proved a formation rule (IX.36) whereby {\displaystyle q(q+1)/2} {\displaystyle q(q+1)/2} is an even perfect number whenever {\displaystyle q} q is a prime of the form {\displaystyle 2^{p}-1} 2^{p}-1 for prime {\displaystyle p} p—what is now called a Mersenne prime. Two millenia later, Euler proved that all even perfect numbers are of this form.This is known as the Euclid–Euler theorem.

Euclid proved that 2p−1(2p − 1) is an even perfect number whenever 2p − 1 is prime (Elements, Prop. IX.36).

For example, the first four perfect numbers are generated by the formula 2p−1(2p − 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.

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