Do Quasiperfect Numbers Really Exist?

[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

 

[Norwegian]

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

 

[dodo]

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).

 

[Joseph Fermat]

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.

 

[blue_raver22]

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.