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A discussion of perfect numbers:

*A positive integer n is said to be perfect if n is equal to the sum of all its positive divisors, not including n itself*

Alternatively, if σ(n) is the sum of all divisors of n, then a perfect number is defined as

σ(n) = 2n

As an example, the number 6 is perfect, since:

σ(6) = 1+2+3+6 = 12 = 2(6)

Indeed, such numbers were known in the time of Euclid and before. The perfect numbers known were:

P1 = 6

P2 = 28

P3 = 496

P4 = 8128

The search for perfect numbers eventually gave rise to the following theorem, which is attributed to Euclid.

*If 2^(k) - 1 is prime, k>1, then n = 2^(k-1)(2^(k)-1) is a perfect number*

(Euler later asserted that all perfect numbers are of this form)

Traditionally, numbers defined as

Mn = 2^(n) - 1

are called Mersenne numbers (or primes) after Marin Mersenne, who showed that the determination of such numbers, and therefore the quest to discover more perfect numbers, can be narrowed to the case where n is prime. There are infinitely many primes, but are there infinitely many Mersenne primes?

From here, many mathematicians searched for larger perfect numbers, and even larger prime numbers. Finding perfect numbers is no doubt one of the oldest problems in number theory. There is no known odd perfect number, probably due to the enormous size of perfect numbers (for example, the perfect number P9 is 22 digits long and P37 is almost 2 million digits long). Clearly we cannot find an odd perfect number in an exhaustive manner.

However, the part that I find interesting is that it has not been proven that an odd perfect number does not or can not exist; although, it has been proven that any

*even*perfect number ends in the digit 6 or the digit 8. But why is it so difficult to show that an odd perfect prime number cannot exist, or why can it not be proven that an odd perfect number may exist? It has not even been shown that there exists infinitely many prime numbers p for which Mp is prime (which in turn *could* show there exists infinitely many perfect numbers). Intuitively one could infer that there must be infinitely many perfect numbers, but it has not been rigorously shown. One of those unsolved problems... but wherein lies the difficulty? Of course this kind of problem is one of curiousity, not usefulness.

pardon my not-so-clear notation. I am not sure how to apply super- and sub-scripts. perhaps if no one can contribute to this discussion, they can at least help me with this