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.

Preprint of "Binary Curves of small fixed genus and gonality with many rational points" on arXiv

A preprint of my paper with Xander Faber, "Binary Curves of small fixed genus and gonality with many rational points," is now available at the arXiv.

"Brazilian Primes Which Are Also Sophie Germain Primes" Appears in Integers

I am pleased that my paper with Hester Graves, "Brazilian Primes Which Are Also Sophie Germain Primes" has appeared in the journal Integers.

This marks my second publication this year, a feat I only equaled in 2014 (which was the last year I published anything). But the year is less than half over, and I have a lot of time to write stuff, so who knows whether I can make it to a third this year. (The length of the refereeing and editing process means probably no.)

This is only the fourth journal I've published in, after Mathematics of Computation, Journal of Number Theory, and the American Mathematical Monthly. I am hoping to expand that with my next few publications.

"Binary Curves of Fixed Genus and Gonality with Many Points" at West Coast Number Theory 2019

I just realized I never posted my slides from WCNT 2019. I spoke about "Binary Curves of Fixed Genus and Gonality with Many Points." It is joint work with Xander Faber. We should have a preprint with updated results coming out soon.

"An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test" appears in Journal of Number Theory

I was just reviewing the entries on this blog and realized I had never posted that "An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test" is appearing in the May 2020 issue of Journal of Number Theory. You can cite it as:

Grantham, Jon. An unconditional improvement to the running time of the quadratic Frobenius test. J. Number Theory 210 (2020), 476--480.