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.

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

A preprint of my paper is here. Keep me in your thoughts as I go through the submission/publication process.

Further results on "Grantham's Problem"

While it is gratifying to see a mathematics paper published, it is sometimes even more gratifying to see that paper cited, because it means that someone cares about the original publication.

A decade ago, I mentioned that someone had addressed a question I asked in a 2001 paper, which he called "Grantham's problem."

Now he and two authors have pushed the computations further, in the paper "Quadratic Frobenius pseudoprimes with respect to x² + 5x + 5".

The results put conditions on a pseudoprime with two prime factors. The known heuristics for the existence of pseudoprimes give ones with many prime factors, so it is not surprising, but it is good to see this evidence.

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

I am about to give a talk on "An Unconditional Improvement to the Running Time of the Quadratic Frobenius Test" at the AMMCS conference.

Talk on Brazilian Primes at SERMON 2019

Here are the slides of the talk I am giving today at the SERMON 2019 conference. It is similar to previous talks I have given on primes that are values of cyclotomic polynomials, but with more emphasis on Brazilian primes, in particular Brazilian Sophie Germain primes.

Moving to the arXiv

When my first journal article was published in 1995, putting reprints on my personal web site seemed very advanced compared to keeping them on a shelf in my office.

Times have changed, however, and arXiv.org seems like a more permanent repository for the reprints than pseudoprime.com. And, frankly, more permanent than some of the journal web sites.

So you can now see all of my reprints (and preprint!) at this arXiv link.

Preprint of "Brazilian Primes Which Are Also Sophie Germain Primes"

A preprint of "Brazilian Primes Which Are Also Sophie Germain Primes" is now available on the math arXiv. In it, Hester Graves and I disprove a conjecture from Schott's 2010 paper on Brazilian primes, namely that no Brazilian primes are also Sophie Germain primes.

What does this mean? A Brazilian prime is a prime number of the form 1+b+b²+...+b^k-1, i.e. a prime whose digits are all 1 when written in base b. (To avoid silliness, you need k>2 and b>1.) For this reason, they are sometimes called "prime repunits".

A Sophie Germain prime is a prime p such that 2*p+1 is also prime.

If you just start computing Brazilian primes, most of them will be of length 3. We show that the length of a Brazilian Sophie Germain prime has to be a prime congruent to 2 mod 3, i.e. 5, 11, 17, 23, etc.

In the paper, we computed all Brazilian Sophie Germain primes up to 10⁴⁴. There are 38,031,404 of them, all but 12 of them of length 5. The 12 exceptions are all of length 11. The smallest one of length 17 is 41969813142886369903423014255641324842178685773056721, which is bigger than 10⁵².

We have actually computed all Brazilian Sophie Germain primes up to 10⁴⁶ (there are 104,890,302 of them) and 10⁴⁸ (we haven't counted them up yet). A later version of the preprint will reflect that.

Submission of the sequence of Brazilian Sophie Germain primes is in progress. A later version of the preprint will also reflect that.

Primes Which Are Values of Cyclotomic Polynomials

In my continuing series of talks on work with Hester Graves on primes which are values of cyclotomic polynomials, I am giving a talk today at the MASON III conference at James Madison University. Here are the slides.

Cyclotomic Goldbach

Last month, at the West Coast Number Theory conference in Chico, CA, I gave a talk on different versions of the classical Goldbach conjecture (and related it to the other Goldbach conjecture I've been talking about recently. Here are the slides.

Talking Again about Another Conjecture of Goldbach

Here are the slides for a talk I am giving tomorrow at the MASON conference in Towson, MD. They are almost identical to the slides for the talk I gave last month at the SERMON conference.

I did, however, get some excellent questions last month that have changed the way I think about the problem, so I hope to incorporate those insights into the talk itself.

Another Conjecture of Goldbach

Here are the slides for a talk I am giving this weekend at the SERMON conference in Johnson City, TN. The basic question addressed is as follows.

Look at all numbers a such that a²+1 is prime. (The sequence starts 1, 2, 4, 6, 10... ) Goldbach conjectured that any number (other than 1) in that sequence is the sum of two previous numbers.  So 2=1+1, 4=2+2, 6=2+4, 10=4+6, etc. We verify this for a up to 2*10¹⁴ and explore how easy it is to find these sums. This is joint work with Hester Graves.

Parallel Computation of Primes of the Form x2+1

Today I gave a talk on Parallel Computation of Primes of the Form x²+1 at Towson University, at the first MASON conference.
You can see my slides here.

I computed all such primes up to 6.25x10²⁸.