Abstract
A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as “Odd Near-Perfect Numbers” and “Deficient-Perfect Numbers”. Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 101500, The total number of prime factors/divisors of an odd perfect number is at least 101, and 108 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.
Highlights
According to Ben Stevens on his paper entitled "A Study on Necessary Conditions for Odd Perfect Numbers", Goto upon using an algorithm, show that 108 is an appropriate lower bound for the largest prime factor of an odd perfect number
The following conclusions were derived from the outcome of the study: (1) Lower Bound for Odd Perfect Numbers has a lower bound of 101500
This is the latest known information on the lower bound of odd perfect numbers
Summary
This study aimed to review the necessary conditions on the existence of an odd perfect number by:
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