A thought on the existence of an odd perfect number

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In summary, the conversation discusses the difficulty of proving the non-existence of an odd perfect number and the known information about it, including the fact that it cannot be a square and the complexity of proofs of non-existence. The conversation also touches on the similarities between mathematics and physics in terms of setting exclusion limits.
  • #1
MathematicalPhysicist
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Well the most obvious approach to prove that such a number doesn't exist is by ad absurdum, or so I think.
Assume there exists an odd perfect number ##2n+1##, then by definition ##2n = \sum_{m\ne 1, 2n+1, m|(2n+1)}m##.

So, since m is odd (since 2n+1 is odd and it divides it), if you can prove that the sum has an odd number of terms then obviously we get a contradiction.
What is known of the number of divisors of an odd number?
 
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The prize is immortality in the mathematical literature.
An odd perfect number cannot be a square. That alone is sufficient to make the sum of true divisors odd, so that's everything you get with that approach.
 
  • #5
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
 
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  • #6
Vanadium 50 said:
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
I prefer to phrase this in a positive way: Proofs of non-existence are much, much harder than proofs of existence. That's why lower bounds in complexity theory (NP=P), and those in number theory, e.g. FLT, are so incredibly complicated. Existence is often very easy: prove there is a group, ring, algebra, vector space: ##\{0\}##. Job done.
 
  • #7
Vanadium 50 said:
This is the great thing about mathematics. An odd perfect number probably doesn't even exist, but we know a lot about it!
It sounds very familiar once you use physics jargon: Mathematicians set many exclusion limits.
 

Related to A thought on the existence of an odd perfect number

1. What is 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). In other words, an odd perfect number is a number that is both abundant and odd.

2. Do odd perfect numbers exist?

As of now, there is no proof that an odd perfect number exists. However, it is also not proven that they do not exist. The search for an odd perfect number is an ongoing area of research in mathematics.

3. How many odd perfect numbers are there?

If odd perfect numbers do exist, it is believed that there is only one. This is based on the fact that there are no known odd perfect numbers and that any potential odd perfect number would have to be incredibly large and rare.

4. What are the implications if an odd perfect number is discovered?

If an odd perfect number is discovered, it would have significant implications in the field of mathematics. It would provide a deeper understanding of number theory and potentially lead to new discoveries in other areas of mathematics.

5. How are scientists searching for odd perfect numbers?

Scientists are using various methods and techniques, such as computer algorithms and mathematical equations, to search for odd perfect numbers. They are also studying the properties of existing perfect numbers to gain insight into the potential existence of odd perfect numbers.

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