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Stream: deprecated: mathematics

Topic: [Blog] Exploration of Perfect, Abundant, and Deficient Nums

Jacob Zelko (May 20 2023 at 16:45):

Hey folks,

I recently wrote a blog post with a very small computational treatment of exploring abundant, deficient, and perfect numbers (the latter only briefly) alongside aliquot sequences. This was my first foray into number theory and had a bit of fun with it! Thought I'd share it in case folks find it interesting: https://jacobzelko.com/05102023043333-perfect-abundant-deficit/

What was interesting to me was some encounters that I experienced while working with, in particular, aliquot sequences. What I did not realize is that an aliquot sequence can grow to be monstrous in terms of computation. To give an example, I was innocently computing aliquot sequences for several abundant numbers and strangely hit errors when computing on the value $n = 138$. I didn't know why I kept getting truncation errors within my computations. I looked up this particular number and discovered that the aliquot sequence for $n= 138$ has $177$ aliquot terms in the sequence! It apparently encounters numbers so big that it was leading to truncation errors.

Otherwise, like I said in the conclusion of my blog post, I don't know what other sorts of computational treatments I could do here and how much analogies hold between math domains, but I would love to hear people's thoughts on that! In terms of computationally specific questions, I was wondering:

1. How can I handle extremely large numbers gracefully?
2. How can I compute upon extremely large numbers gracefully?

What I mean by huge for example is the $8^{\text{th}}$ perfect number, $2305843008139952128$, and so forth.

Thanks all!

~ jz

David Egolf (May 20 2023 at 17:29):

I enjoyed reading your post. It was great to learn a little more Julia, and fun to see your numerical visualizations! Regarding working with large numbers, I wonder if this could be relevant to you: https://docs.julialang.org/en/v1/base/numbers/#BigFloats-and-BigInts

I know very little about number theory. But I wonder if the concepts of "deficient", "perfect" and "abundant" generalize to certain kinds of categories. I suppose what we would need is:

• a way to say if one object is a divisor of another object
• a way to add objects
• a way to compare the size of objects

If we have the three things above available to us, then we can describe an object $A$ as "abundant" if the sum of its divisor objects is larger than $A$. Now I'm a bit curious - what are some examples of settings besides the integers that provide the three things listed above?

David Egolf (May 20 2023 at 17:42):

Here's an initial idea for "categorifying" the above list of three requirements, in a category with coproducts:

• Say that $A$ divides $B$ if the coproduct of $A$ with itself some finite number of times is isomorphic to $B$
• Let the sum of two objects be their coproduct (so the sum is defined up to isomorphism)
• Say that $A \leq B$ if there is a monomorphism from $A$ to $B$

Applying this to the category of finite sets I think gives us something similar to the usual notions of divisiblity, addition, and ordering for the natural numbers.

Another approach for divisibility might be, in a category with products:

• Say that $A$ divides $B$ if there exists some $C$ so that the product of $A$ and $C$ is isomorphic to $B$

John Baez (May 20 2023 at 18:08):

Both these approaches work, in the sense that starting from the category of finite sets and functions we get the usual concept of divisibility for natural numbers.

Related: for any category $\mathsf{C}$ with finite products we can create a preorder where we say objects $a,b \in \mathsf{C}$ have $b \le a$ iff for some $c \in \mathsf{C}$ we have $a \cong b \times c$.

John Baez (May 20 2023 at 18:09):

In this case we can say $a$ is divisible by $b$.

John Baez (May 20 2023 at 18:10):

We could also do this for a monoidal category by saying $b \le a$ iff $\exists c \; a \cong b \otimes c$.

John Baez (May 20 2023 at 18:11):

For example with finite-dimensional vector spaces and the usual tensor product this gives a preorder equivalent to the natural numbers with their usual partial ordering by divisibility. One little thing to beware of: with this partial ordering on natural numbers, the biggest natural number is 0.

John Baez (May 20 2023 at 18:12):

For example $1 \le 2 \le 6 \le 24 \le 120 \le 720 \le \cdots \le 0$

Jacob Zelko (May 20 2023 at 21:08):

Hey @David Egolf! Thanks for having a chance to read through the post! Just responding to some various thoughts you raised:

I enjoyed reading your post. It was great to learn a little more Julia, and fun to see your numerical visualizations! Regarding working with large numbers, I wonder if this could be relevant to you: https://docs.julialang.org/en/v1/base/numbers/#BigFloats-and-BigInts

Glad you enjoyed the Julia bit but in the post, I personally wasn't so worried about the implementation in any specific language but more so how working with large numbers generally works in computational mathematics. However, for the specific implementation that reference looks super helpful -- will give it a look.

I really enjoyed the idea of categorifying the number species, but my question here after this is what would this allow us to say about these numbers and what insights could it lead to? I may be too strongly wearing my computational hat here, but I am curious what you all are thinking as next steps here after this "categorification"?

David Egolf (May 20 2023 at 21:27):

Jacob Zelko said:

I really enjoyed the idea of categorifying the number species, but my question here after this is what would this allow us to say about these numbers and what insights could it lead to? I may be too strongly wearing my computational hat here, but I am curious what you all are thinking as next steps here after this "categorification"?

I find it can be fun and interesting to take a math concept and see if I can try to express some aspects of it in categorical language. Sometimes this can help me notice parallels between related concepts in different categories, too. It's also good practice for me, as I try to get more comfortable with the language of category theory. So, I didn't have any specific insights or next steps in mind. I just thought it was interesting and fun to try and connect these ideas to category theory.

That being said, if I were to learn more about perfect/abundant/deficient numbers, one of my first questions would probably be: How much in common do the numbers that fall into one of these classes have? Are there any interesting theorems that apply to any abundant number, for example? If so, can any of these theorems be translated to a more general setting (as discussed above)? If not, why not?

David Egolf (May 20 2023 at 21:46):

For example, wikipedia says that every multiple of a perfect number (excluding the perfect number itself) is abundant. Proving that might be interesting. It might also be interesting to check if "every multiple of a perfect object (except for the perfect object itself) is an abundant object" holds in a more general setting, using one of the notions of divisibility described above.