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.