My name is Wouter van Doorn, I am 32 years old and I live in the Netherlands. In 2010 I managed to solve a question about sums of unit fractions (unit fractions are fractions like 1/2 and 1/17, which have 1 as a numerator) that was asked by Paul Erdős and Ronald Graham in the 1970s. Ever since I have been exploring this subject further, trying to answer related questions and writing up an article. It is now 12 years later and I am still not finished, although I should be quite close. In any case, I have always felt that a written document is not the right way to convey and explain my ideas to other people. I think that videos in which I explain the proofs and theorems would be a way better medium to illustrate everything. The only problem is that the paper I have written so far is over 60 pages of fairly dense mathematics, so it is going to take a lot of hours to convert all of this into video format. But SoME2 might be just the right motivation to start!
Quick Summary
To explain the topic a little, let's do a bit of mathematics. We choose a positive integer A, let's say A = 2, and then look at 1/2, 1/2 + 1/3, 1/2 + 1/3 + 1/4, etc. We write such a sum as one fraction and then we are interested in when the denominator of the resulting fraction decreases. For A = 2 we get the following sums and you should focus in particular on the denominator of the fraction on the right-hand side:
As you can see, the denominator initially grows from 2, to 6, to 12, to 60, but then, when the sum reaches 1/6, it decreases to 20. What I am interested in is the following question: for a given positive integer A, if we start writing those sums starting at 1/A, when does the denominator decrease?
For example, if we denote by b(A) the smallest integer larger than A such that 1/A + .. + 1/b(A) has a smaller denominator than 1/A + .. + 1/(b(A) - 1), then I can prove that b(A) < 4.38A for A > 4, whereas b(A) > A + 0.54 * log(A) for all A that are large enough.
The goal in the collaboration is to prove these bounds and much more in such a way that an enthusiastic undergraduate student with some basic number theory knowledge should be able to understand almost everything, potentially barring a few theorems that we will accept as black boxes.
Target medium
The goal is to make a video series where I explain step-by-step the entire paper I have written. This is why I am looking for a very patient producer/editor. Since this is not going to be one video but an entire series, the end result will definitely not be suited for SoME2, but this project could be the impetus for it. Since this may all sound a bit scary, there is also some good news:
First of all, the videos do not have to be anywhere near 3b1b level of editing. I am more thinking the style of Michael Penn. If you are not familiar with him, here is his most recent video, to give an example. It will just be me and a whiteboard with some number theory on it.
Secondly, you do not have do be well-versed in number theory to be able to help me. Even if you can't quite remember the definition of a prime number, all I need is the video editing. I will do the mathematics myself. If you however do know some stuff about numbers, that is also allowed!
Thirdly, even if the video series might not be very well suited for SoME2, it is still possibly to make one separate sort of introductory video that can be submitted to SoME2! I would love this actually.
Finally - and I understand that this is not wholly in the spirit of SoME2- I am also willing to pay a (small) amount of money to someone willing to help me produce this, since the length of the project will go well beyond the summer of 2022.
More details
If you are interested in any way, you can find an introductory reddit post about the lower bound b(A) > A + 0.54 * log(A) here. My paper can be found here.
Contact details
If you have any questions, suggestions or interested in collaborating, you can either respond to this topic, you can send me a message via reddit, or you can send an e-mail to wonterman1@hotmail.com.
About the author
My name is Wouter van Doorn, I am 32 years old and I live in the Netherlands. In 2010 I managed to solve a question about sums of unit fractions (unit fractions are fractions like 1/2 and 1/17, which have 1 as a numerator) that was asked by Paul Erdős and Ronald Graham in the 1970s. Ever since I have been exploring this subject further, trying to answer related questions and writing up an article. It is now 12 years later and I am still not finished, although I should be quite close. In any case, I have always felt that a written document is not the right way to convey and explain my ideas to other people. I think that videos in which I explain the proofs and theorems would be a way better medium to illustrate everything. The only problem is that the paper I have written so far is over 60 pages of fairly dense mathematics, so it is going to take a lot of hours to convert all of this into video format. But SoME2 might be just the right motivation to start!
Quick Summary
To explain the topic a little, let's do a bit of mathematics. We choose a positive integer A, let's say A = 2, and then look at 1/2, 1/2 + 1/3, 1/2 + 1/3 + 1/4, etc. We write such a sum as one fraction and then we are interested in when the denominator of the resulting fraction decreases. For A = 2 we get the following sums and you should focus in particular on the denominator of the fraction on the right-hand side:
1/2 = 1/2 1/2 + 1/3 = 5/6 1/2 + 1/3 + 1/4 = 13/12 1/2 + 1/3 + 1/4 + 1/5 = 77/60 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 29/20
As you can see, the denominator initially grows from 2, to 6, to 12, to 60, but then, when the sum reaches 1/6, it decreases to 20. What I am interested in is the following question: for a given positive integer A, if we start writing those sums starting at 1/A, when does the denominator decrease?
For example, if we denote by b(A) the smallest integer larger than A such that 1/A + .. + 1/b(A) has a smaller denominator than 1/A + .. + 1/(b(A) - 1), then I can prove that b(A) < 4.38A for A > 4, whereas b(A) > A + 0.54 * log(A) for all A that are large enough.
The goal in the collaboration is to prove these bounds and much more in such a way that an enthusiastic undergraduate student with some basic number theory knowledge should be able to understand almost everything, potentially barring a few theorems that we will accept as black boxes.
Target medium
The goal is to make a video series where I explain step-by-step the entire paper I have written. This is why I am looking for a very patient producer/editor. Since this is not going to be one video but an entire series, the end result will definitely not be suited for SoME2, but this project could be the impetus for it. Since this may all sound a bit scary, there is also some good news:
First of all, the videos do not have to be anywhere near 3b1b level of editing. I am more thinking the style of Michael Penn. If you are not familiar with him, here is his most recent video, to give an example. It will just be me and a whiteboard with some number theory on it.
Secondly, you do not have do be well-versed in number theory to be able to help me. Even if you can't quite remember the definition of a prime number, all I need is the video editing. I will do the mathematics myself. If you however do know some stuff about numbers, that is also allowed!
Thirdly, even if the video series might not be very well suited for SoME2, it is still possibly to make one separate sort of introductory video that can be submitted to SoME2! I would love this actually.
Finally - and I understand that this is not wholly in the spirit of SoME2- I am also willing to pay a (small) amount of money to someone willing to help me produce this, since the length of the project will go well beyond the summer of 2022.
More details
If you are interested in any way, you can find an introductory reddit post about the lower bound b(A) > A + 0.54 * log(A) here. My paper can be found here.
Contact details
If you have any questions, suggestions or interested in collaborating, you can either respond to this topic, you can send me a message via reddit, or you can send an e-mail to wonterman1@hotmail.com.