| dc.description.abstract | This study delves into the realms of number theory, specifically investigating perfect
numbers and partitions of odd numbers. Perfect numbers, which are integers equal to the
sum of their proper divisors, excluding themselves, have intrigued mathematicians for
centuries. While it is established that even perfect numbers can be expressed as 2p-1(2p -1)
, where 𝑝 and 2p -1 are prime numbers (Mersenne primes), the existence of odd perfect
numbers remains an unsolved problem. The first part of the thesis explores perfect
numbers, tracing their history from ancient Greek mathematicians to modern scholars, and
discussing various results and conjectures. The focus then shifts to partitions of odd
numbers, which represent different ways of expressing an odd number as a sum of positive
integers. The study utilizes an algorithms that has demonstrated that a positive even integer
can be partitioned into all pairs of odd numbers. Using this approach, it is shown that any
positive odd number 2𝑛 + 1 can be partitioned into all pairs of both odd and even numbers
and from the set of these partitions, it is shown that there exist a proper subset containing
all proper divisors of 2𝑛 + 1. Using the results from the partitions and the facts that there
exist infinitely many odd numbers, it’s therefore verifies that odd perfect numbers do not
exist. | en_US |
