# Do Quasiperfect Numbers Really Exist?

• Joseph Fermat
In summary, the conversation discusses the concept of Quasiperfect numbers and the difficulty in proving their existence. Joseph presents a proof that shows the impossibility of Quasiperfect numbers using a transformation process. However, another participant points out a flaw in the proof and suggests seeking new ideas to continue the search for Quasiperfect numbers.
Joseph Fermat
A Quasiperfect number is any number for which the sum of it's divisors is equal to one minus twice the number, or a number where the following form is true,

σ(n)=2n+1

One of the well known and most difficult questions in mathematics is whether such numbers exist at all. I have created a rather interesting proof to show that quasiperfect numbers do not exist. I use a process of transformation to create a situation necessary for the existence of a quasiperfect number, and then show that such a situation is impossible, therefore disproving the possibility of a quasiperfect number.

View attachment On the Nonexistence of Quasiperfect Numbers.pdf

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Why not use the same argument with n=2x+1 to prove that odd numbers do not exist?

Hi, Joseph,
there is a problem when going from eq.8 to eq.9: $1 - (h(n) - 2)$ is not $-(h(n)+1)$ (which is negative), but $3 - h(n)$ (which is positive).

Dodo said:
Hi, Joseph,
there is a problem when going from eq.8 to eq.9: $1 - (h(n) - 2)$ is not $-(h(n)+1)$ (which is negative), but $3 - h(n)$ (which is positive).
Which would mean that my proof is fallous. Oh, well back to the drawing board. Anyone have any ideas where to go from here. Any help would be appreciated.

Thank you for sharing your proof on quasiperfect numbers. Your approach of using transformation to create a necessary situation for the existence of a quasiperfect number is interesting. However, it is important to note that the existence of quasiperfect numbers is still an open question in mathematics and has not been definitively proven or disproven.

While your proof may provide evidence against the existence of quasiperfect numbers, it is always important to consider alternative perspectives and continue to explore different approaches to this problem. As scientists, it is our duty to continue to question and seek answers, even if they may challenge our current understanding.

Additionally, it is worth mentioning that even if quasiperfect numbers do not exist, your proof and approach can still contribute to the understanding of number theory and lead to further discoveries in this field. Thank you for your contribution to this ongoing mathematical inquiry.

## 1. What are quasiperfect numbers?

Quasiperfect numbers are positive integers that have the sum of their proper divisors equal to a power of 2. In other words, the sum of all the positive divisors of a quasiperfect number (excluding the number itself) is equal to 2 raised to some integer power.

## 2. How are quasiperfect numbers different from perfect numbers?

Unlike perfect numbers, which have the sum of their proper divisors equal to the number itself, quasiperfect numbers have their sum of proper divisors equal to a power of 2. Additionally, there are only a few known quasiperfect numbers, while there are infinitely many known perfect numbers.

## 3. What is the significance of quasiperfect numbers?

Quasiperfect numbers have been studied in number theory and have been found to have interesting connections to other areas of mathematics, such as algebraic geometry and modular forms. They also have applications in coding theory and cryptography.

## 4. How are quasiperfect numbers proven to be valid?

To prove that a number is quasiperfect, one must show that the sum of its proper divisors is equal to a power of 2. This can be done through various mathematical techniques, such as factoring the number or using number theoretic theorems and properties.

## 5. Are there any unsolved problems related to quasiperfect numbers?

Yes, there are still many open questions and conjectures related to quasiperfect numbers. For example, it is not known if there are any odd quasiperfect numbers, or if there are infinitely many quasiperfect numbers. Additionally, it is still an open problem to find an efficient algorithm for determining whether a given number is quasiperfect.

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