Publication of "On the maximum gonality of a curve over a finite field"

My paper with Xander Faber and Everett Howe, "On the maximum gonality of a curve over a finite field" appeared in Algebra & Number Theory earlier this year. This is probably the fanciest journal my work has appeared in, though I owe most of the fanciness to my co-authors.

Publication of "No New Goormaghtigh Primes up to 10^700"

 

My paper, "No New Goormaghtigh Primes up to 10⁷⁰⁰" appeared in Integers last year.

This was a super-fun paper to write. I had not heard of the Goormaghtigh Conjecture until a virtual talk I attended during the pandemic. I stopped listening 10 minute in when the speaker moved on to generalizations and had a hard time stopping thinking about the topic.

The conjecture is that the only two numbers which are repunits (all 1s) in two different bases are 31 and 8191. I came up with the idea of restricting to prime solutions and was able to show that those are the only two such numbers up to 10⁷⁰⁰.

Publication of "Fibonacci primes, primes of the form 2^n-k and beyond"

 

Last year, my paper with Andrew Granville, "Fibonacci primes, primes of the form 2^n-k and beyond" appeared in the Journal of Number Theory.

Publication of "Ternary and quaternary curves of small fixed genus and gonality with many rational points" in Experimental Mathematics

I haven't kept up this blog in a couple of years, and I've had a few articles appear since then. Let's try to get caught up.

My paper with Xander Faber, "Ternary and quaternary curves of small fixed genus and gonality with many rational points" appeared in Experimental Mathematics in 2023.

Publication of "Finding a widely digitally delicate prime" in Integers

 

My paper "Finding a widely digitally delicate prime" appeared in the journal Integers earlier this year.

Publication of "On Integers Whose Sum Is the Reverse of Their Product" in Fibonacci Quarterly

 

My paper with Xander Faber, "On Integers Whose Sum Is the Reverse of Their Product" has appeared in the latest issue of Fibonacci Quarterly. It is my 13th published paper.

A freely-accessible version is available at the arXiv.

Preprint of "On the Maximum Gonality of a Curve over a Finite Field" on the math arXiv

 

A preprint of my paper with Xander Faber and Everett Howe, "On the Maximum Gonality of a Curve over a Finite Field" has appeared on the math arXiv. This is the conclusion to the "Gonality Trilogy".

"Binary curves of small fixed genus and gonality with many rational points" appears in Journal of Algebra

 

My paper with Xander Faber, "Binary curves of small fixed genus and gonality with many rational points" appeared in the 1 May 2022 issue of the Journal of Algebra. This is my 12th publication (although its print publication precedes our ternary/quaternary paper, so I will have to renumber at some point).

Preprint of "Only finitely many s-Cullen numbers are repunits for a fixed s≥2" on the math arXiv

A preprint of my paper with Michael Filaseta and Hester Graves, "Only finitely many s-Cullen numbers are repunits for a fixed s≥2" has appeared on the math arXiv. This is a sequel to an earlier paper with Hester, which proved that there were only finitely many solutions for all s, subject to the abc conjecture.

This proof is unconditional. I am now working on computations to extend 65536 to a much larger number -- it should be possible to get it into the millions.

"Proof of Two Conjectures of Andrica and Bagdasar" Appears in Integers

My paper, "Proof of Two Conjectures of Andrica and Bagdasar" has appeared in the journal Integers. It uses techniques from my earlier work to show that certain families of pseudoprimes are infinite. I hope that others find this technique useful.

Preprint of "Finding a Widely Digitally Delicate Prime" available on the math arXiv

In September, I posted a preprint of "Finding a Widely Digitally Delicate Prime" on the math arXiv. It may not end up published in this form, but because it described the only known example of such a prime, and how I found it, I wanted to have it documented.

"Ternary and Quaternary Curves of Small Fixed Genus and Gonality With Many Rational Points" Appears Online in Experimental Mathematics

 

My paper with Xander Faber, "Ternary and Quaternary Curves of Small Fixed Genus and Gonality With Many Rational Points," has appeared in Experimental Mathematics online. Based on my estimate of the journal's backlog, I expect print publication in 2023.

This is my tenth journal article published, although I may have to re-do the numbering in 2023.

"The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits" Appears in the Journal of Integer Sequences

 

Catching up from earlier this year, my paper with Hester Graves, "The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits" appeared in the Journal of Integer Sequences.

This is my ninth journal article published.

Preprint of "On Integers Whose Sum is the Reverse of their Product" available on the arXiv

 

The preprint of "On Integers Whose Sum is the Reverse of their Product", feat. Xander Faber is now available on the math arXiv.

Have you ever noticed that 9+9=18, while 9x9=81, which is 18 backwards? My co-author's kids did. The question of what other pairs of numbers a and b have a+b equal to axb backwards led us to write this paper. We give a technique for solving it in all bases. We have seen a list of these numbers in base 10 on the web, but we are aware of no other proof.

"A recent conjecture by Faber-Grantham"

What's even better than writing a paper? Having someone else write one answering a question you asked. In "Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points," Xander Faber and I conjectured:

In a recent preprint, Floris Vermeulen proved this conjecture.

Preprint of "Ternary and Quaternary Curves of Small Fixed Genus and Gonality with Many Rational Points" available on the arXiv

 Xander Faber and I have made the second paper in our Gonality Trilogy available on the math arXiv.

Video of "An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test"

A video of my talk on "An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test" at the recent PAJAMAS conference is available on YouTube.

Since I am likely to have more talks available online in the future, I have created a talks page listing them all (at this moment, listing them both).

While I was at it, I created a papers page as well. My previous one had not been updated since 2014, and did not have great links. This one links to both the arXiv copies and the journals (for those which have been published).

Preprint of "The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits" on the arXiv

A preprint of "The abc Conjecture Implies That Only Finitely Many Cullen Numbers Are Repunits" is now available on the math arXiv. In it, Hester Graves and I show that if you assume the abc Conjecture, you can show that finitely many Cullen numbers are repunits to any base. (Sort of like it says in the title.)

Frequently Asked Questions (or so I imagine):

1. What is the abc conjecture? It says that if you have integers a, b, and c with a+b=c, the product of all primes dividing a, b and c is almost never much smaller than c. For more, you can read the Wikipedia article.
2. What are Cullen numbers? Cullen numbers are numbers of the form n2ⁿ+1, for a positive integer n.
3. What are repunits? They are numbers all of whose digits are 1. So 111 or 1111. But! Those are base-10, and in this paper, we consider any base, so 7 is a repunit, because it is 111 in base 2. 2801 is a repunit, because it is 1111 in base 7. We don't allow 11, because every number has a base where it's 11.
4. Does your result hold for s-Cullen numbers? Yes. For any given s, there are finitely many s-Cullen numbers that are repunits (assuming the abc conjecture).
5. What are s-Cullen numbers? Numbers of the form nsⁿ+1.
6. When did you start using the abc conjecture? Just now! Hester had this cool idea, and it was fun to see how it worked out.
   

Video of "Binary and Ternary Curves of Fixed Genus and Gonality with Many Points"

I am happy to say that the video of my talk at NTOC 2020, "Binary and Ternary Curves of Fixed Genus and Gonality with Many Points" is available for viewing online. (This represents joint work with Xander Faber.) As far as I can tell, this is the first talk of mine to be publicly available for viewing. I am hoping it is the first of many!

"Binary and Ternary Curves of Fixed Genus and Gonality with Many Points" at Number Theory Online Conference

Here are the slides from a talk I just gave at the Number Theory Online Conference.