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February 2nd, 2017, 09:38 PM   #1
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Perfect Numbers

I was searching some mathematical formulas and come across this question. Can any one help me to solve this.?

I studied that a perfect number is one which equals the sum of its proper divisors.

For example: the number 18 is abundant since

18 is less than 1 + 2 +3 + 6 + 9 =21,
the number 15 is deficient since 15 is greater than 1+3+5 =9 and

6 =1+ 2 + 3 is perfect.

Then I found the equation to find perfect even numbers : m = 2^(p-1) * (2^p - 1) , where p is prime number.

Now the main question is from this equation : Can we find how many even & odd perfect numbers are there?
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February 2nd, 2017, 09:46 PM   #2
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It's an open question as to whether any odd perfect numbers exist. This is a famous problem in math.

Here's an article about it. They've checked up to $10^{300}$ and not found any. That's $1$ with $300$ zeros. But compared to all the natural numbers that there are, it's just a drop in the bucket.

Odd Perfect Numbers: Do They Exist?
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February 3rd, 2017, 04:53 AM   #3
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I think this is still debatable questions, still I have found one article which can help you to solve this:

Odd Perfect Number -- from Wolfram MathWorld
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February 4th, 2017, 03:06 PM   #4
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Quote:
Originally Posted by SophiaRivera007 View Post
I think this is still debatable questions, still I have found one article which can help you to solve this:

Odd Perfect Number -- from Wolfram MathWorld
Hey My Math Forum members and guests!

Did you see the credit given at the bottom of this wolfram link for contributing information? Our very own CRGreathouse!!!

What a wonderful surprise and delight
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February 5th, 2017, 02:47 PM   #5
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Quote:
Originally Posted by AshBox View Post
I was searching some mathematical formulas and come across this question. Can any one help me to solve this.?

I studied that a perfect number is one which equals the sum of its proper divisors.

For example: the number 18 is abundant since

18 is less than 1 + 2 +3 + 6 + 9 =21,
the number 15 is deficient since 15 is greater than 1+3+5 =9 and

6 =1+ 2 + 3 is perfect.

Then I found the equation to find perfect even numbers : m = 2^(p-1) * (2^p - 1) , where p is prime number.

Now the main question is from this equation : Can we find how many even & odd perfect numbers are there?
Does the equation (m=....) generate all even perfect numbers?
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February 5th, 2017, 03:28 PM   #6
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Yes. Note that 2^p - 1 needs to be prime as well for m to be a perfect number. It's not known whether the number of perfect numbers is finite.
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February 5th, 2017, 10:33 PM   #7
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Perfect Numbers

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Originally Posted by mathman View Post
Does the equation (m=....) generate all even perfect numbers?
Yes, absolutely this works and provide you perfect numbers.
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