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Invitation to Modular Forms: Representation of an integer into sum of eight squares

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Tags: number-theoryalgorithmsmodular-forms

The goal of this article is to invite algorithmists (and competitive programmers) to the fascinating theory of modular forms, which is a very deep tool in number theory. Here we consider the computational problem of counting the number of ways to represent an integer into sum of eight squares of integers.

https://sirawit.notion.site/Invitation-to-Modular-Forms-Representation-of-an-integer-into-sum-of-eight-squares-2589cafc3f3380148daee16c13802477

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5 Overall score*

54 Rank

12 Votes

7 Comments

Comments

I like the idea, but it’s a bit like drinking from a firehose.

I enjoyed the opening a lot, where you discussed the sum-of-eight-squares problem, giving the formulas and code for how you would compute it efficiently.

But then the mathematics quickly became very dense and a very fast pace. There were so many definitions back to back. I understood the definition of a modular form, but not why anyone would want to use it, and you didn’t give an example of it with the definition. And then I realised that the material in the article covered a few lectures and it would take me hours to read and understand it so I stopped.

It’s hard for me to give a rating since it would probably be a really useful resource to someone who had come across modular forms in their studies and wanted to learn more about them.

8.3

If you enjoy really getting into the math this is for you.

8.2

This is excellent work with the proofs provided to get to an O(sqrt(n)) runtime for the the number of ways tuples in Z8 can be written. That being said, perhaps a preface on some prereq knowledge (given this is labelled for undergrads as well) in abstract algebra (e.g. def’ns symmetric groups, holomorphic. memorphic, modular forms, et al), complex analysis,…

The Ranking score you received from my review is an average of these individual scores: Motivation: 9.9 Clarity: 7.7 Novelty: 8 Memorability: 7.25

This website/representation has performance issues, even though most of it is static, and makes reading it difficult. I lost patience for the website after the first part because this was too distracting.

I think the topic is great and it’s clear that you’ve put a lot of effort into this. However I’m afraid it might be a bit out of scope for the SOME. Even for professional mathematicians this is a difficult and complex read. It feels like this material would be almost impossible for a broad audience to understand. It is not your problem — most of the books and lecture notes on modular forms have exactly the same problem, but generally speaking selection of the good topic is 75% of the success.

The explanations could be distilled into the key points. The latter half of the article reads a bit too much like lecture notes for it to be especially memorable.