By a near perfect number we mean a number n whose sum of positive divisors excluding n is equal to n-1. For example, 16 is near perfect number.
It is possible to use proximity space theory to classify near perfect numbers. Here are two possibilities.
way one: Given a set of numbers X endowed with an Efremovic proximity relation, let
A be set of perfect numbers and let x be a number in X. Then
\[
x\ \mbox{is close to}\ A,\ \mbox{provided}\ d(x,A) \leq \varepsilon
\]
where d(x,A) is the Hausdorff distance between x and A, i.e.,
\[
d(x,A) = \left\{d(x,a): a\in A\right\},
\]
where d(x,a) is the standard distance between x and perfect number a. And $\varepsilon$ is a small positive number that you choose.
way two: Given a set of numbers X endowed with a descriptive Efremovic proximity relation, let A be set of perfect numbers and let x be a number in X and let $\Phi$ be a set of probe functions that represent features of the numbers. Then compute the similarity distance between a candidate x and the set of perfect numbers A.
See my paper on pattern discovery to learn about similarity distance.
It is possible to use proximity space theory to classify near perfect numbers. Here are two possibilities.
way one: Given a set of numbers X endowed with an Efremovic proximity relation, let
A be set of perfect numbers and let x be a number in X. Then
\[
x\ \mbox{is close to}\ A,\ \mbox{provided}\ d(x,A) \leq \varepsilon
\]
where d(x,A) is the Hausdorff distance between x and A, i.e.,
\[
d(x,A) = \left\{d(x,a): a\in A\right\},
\]
where d(x,a) is the standard distance between x and perfect number a. And $\varepsilon$ is a small positive number that you choose.
way two: Given a set of numbers X endowed with a descriptive Efremovic proximity relation, let A be set of perfect numbers and let x be a number in X and let $\Phi$ be a set of probe functions that represent features of the numbers. Then compute the similarity distance between a candidate x and the set of perfect numbers A.
See my paper on pattern discovery to learn about similarity distance.
Patrick,
The descriptive proximity space approach holds the most promise in solving the original as well the problem you mention. Here is how the approach works.
1. Let $\Phi$ be a set of probe functions representing features of the numbers we are interested in classifying. A probe function $\phi\in \Phi$ is a mapping
\[
\phi: \mathbb{R}\rightarrow \mathbb{R}\ \mbox{that extracts a feature value from a number}.
\]
2. Choose a perfect number $m$ that serves as a pattern generator.
3. Find all numbers that are descriptively close to $m$ and obtain the pattern
$P_{\Phi}(m)$.
Dear Prof. James Peters
Do you have another proof other using proximity?
Jaber,
Yes, it is possible to use descriptive topology to classify perfect numbers. Try the following approach.
1. Let X be a descriptive topological space. A descriptive topology is a family of sets $\tau_{\Phi}$ in X such that the intersection and union properties are satisfied and every x in X has a description. For example, features of x will be either even or odd integer.
2. Let A be a perfect numbers in X such that each x in A either even or odd and x is a near perfect number.
3. Find B in X such that B contains A and every number $b$ is a perfect number that is desriptively close to one of the near perfect numbers in A, provided the descriptive distance between some $a$ in A and some $b$ in B is zero.
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