Why an odd perfect number, if exists, is not divisble by 3?

[l-1j-cho]

So a number of mathematicans showed that an odd perfect number is not divisble by either 3, 5,or 7. I tried really hard to find a proof for that, but I haven't succeeded to do so.
Could anyone please help me to find a proof that an odd perfect nmber is not divisble by 3?

 

[l-1j-cho]

never mind. it might
be divisible by 3, 5, or 7

 

[mathwonk]

the result i recall is that an odd perfect number has at least 7 prime factors. that result is some 40 years old, and due i believe to carl pomerance.

 

[mathwonk]

according to wiki the lower bound is now up to 9 distinct prime factors.

 

[blue_raver22]

I would approach this question by first understanding the concept of an odd perfect number. An odd perfect number is a positive integer that is equal to the sum of its proper positive divisors (excluding itself). This means that an odd perfect number must have an even number of divisors, as it includes 1 and itself, and all other divisors come in pairs.

Now, let us consider the prime factorization of an odd perfect number. If we assume that an odd perfect number is divisible by 3, then it can be written as 3^k * m, where k is a positive integer and m is some other positive integer. This means that the sum of the divisors of this number would be (1 + 3 + 3^2 + ... + 3^k)(1 + m), which is an odd number (since the first part of the sum is always odd). However, an odd perfect number must have an even number of divisors, so this contradicts our initial assumption that it is divisible by 3.

In simpler terms, an odd perfect number cannot be divisible by 3 because it would result in an odd number of divisors, which is not possible for an odd perfect number.

Furthermore, it has been proven by mathematicians that an odd perfect number must have at least one prime factor greater than 10. Since 3 is the smallest prime number greater than 10, it follows that an odd perfect number cannot be divisible by 3.

In conclusion, we can say that an odd perfect number, if it exists, cannot be divisible by 3 because it would result in an odd number of divisors, which contradicts the definition of an odd perfect number. Additionally, the fact that an odd perfect number must have at least one prime factor greater than 10 further supports this conclusion.