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

  • Thread starter l-1j-cho
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In summary, a group of mathematicians have proven that an odd perfect number cannot be divided by 3, 5, or 7. The speaker has tried but failed to find a proof for this, and is asking for help. However, there is a 40-year-old result that states an odd perfect number must have at least 7 prime factors, and this has now been updated to a lower bound of 9 distinct prime factors.
  • #1
l-1j-cho
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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?
 
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  • #2
never mind. it might
be divisible by 3, 5, or 7
 
  • #3
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.
 
  • #4
according to wiki the lower bound is now up to 9 distinct prime factors.
 
  • #5


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.
 

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

1. Why are odd perfect numbers not divisible by 3?

This is because if an odd perfect number exists, it must have a special form known as the Eulerian form, which is 12k + 1 for some positive integer k. Since 3 is not a factor of 12, it cannot divide an odd perfect number in this form.

2. Is there a mathematical proof for odd perfect numbers not being divisible by 3?

Yes, there is a well-known proof by Leonhard Euler which shows that if an odd perfect number exists, it must be of the form 12k + 1 and therefore not divisible by 3.

3. Can an odd perfect number be divisible by any other numbers?

It is possible for an odd perfect number to be divisible by other numbers, such as 5 or 7, but it cannot be divisible by all prime numbers. This is because if it were divisible by all primes, it would be divisible by their product, which would make it a perfect number, contradicting its oddness.

4. Are there any known odd perfect numbers that are not divisible by 3?

No, there are no known odd perfect numbers. It is still an open question whether they exist or not. However, based on Euler's proof, it is clear that if an odd perfect number exists, it cannot be divisible by 3.

5. What would be the significance of finding an odd perfect number that is not divisible by 3?

If an odd perfect number were to be discovered that is not divisible by 3, it would provide new insights and challenges for mathematicians in understanding the properties and structure of perfect numbers. It could also potentially open up new avenues in number theory and contribute to our understanding of the fundamental principles of mathematics.

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