Monday, October 19, 2020
Wednesday, September 23, 2020
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).
Wednesday, September 09, 2020
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):
- 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.
- What are Cullen numbers? Cullen numbers are numbers of the form n2n+1, for a positive integer n.
- 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.
- 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).
- What are s-Cullen numbers? Numbers of the form nsn+1.
- 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.
Wednesday, September 02, 2020
Wednesday, June 03, 2020
Tuesday, May 26, 2020
Sunday, May 10, 2020
Monday, May 04, 2020
Monday, March 16, 2020
"An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test" appears in Journal of Number Theory
Grantham, Jon. An unconditional improvement to the running time of the quadratic Frobenius test. J. Number Theory 210 (2020), 476--480.