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.

Talk at Pomerance Conference

Slides for my talk from June's Pomerance conference are now available at the conference website.

All My Reprints

Inspired by the availability of a reprint for my latest publication, I have updated my list of reprints. So 2 papers this year, but 6 in total over the past 20 years! I have two projects in the computations-in-progress-but-not-yet-written-up stage, so hopefully I'll end up somewhere between the two over the next few years.

Full Text of "Repeatedly Appending Any Digit to Generate Composite Numbers"

The American Mathematical Monthly recently e-mailed my co-authors and me a PDF copy of our article. The e-mail contained the line, "You may post it on www.arXiv.org and your personal website if you so choose." Very reasonable!

It is available at http://www.pseudoprime.com/amer.math.monthly.121.05.416-wagon.pdf.

So you can read it even if you don't otherwise have access to the Monthly, which Wikipedia tells me is the "most widely read mathematics journal in the world."

Publication of "Repeatedly Appending Any Digit to Generate Composite Numbers"

I'm proud to say that "Repeatedly Appending Any Digit to Generate Composite Numbers," a paper I co-authored with Witold Jarnicki, John Rickert and Stan Wagon, has appeared in the May 2014 American Mathematical Monthly. If you are an MAA member (which, um, I'm not), access it through their web site.

Print Publication of Constructing Carmichael numbers through improved subset-product algorithms

"Constructing Carmichael numbers through improved subset-product algorithms," co-authored with the late Red Alford, as well as Steven Hayman and Andrew Shallue, published online last summer, has been placed in the March 2014 issue of Mathematics of Computation.

That means if you want to cite it, you can now cite it as:
Constructing Carmichael numbers through improved subset-product algorithms. Math. Comp. 83 (2014), no. 286, 899-915.

I'm still waiting for it to appear in MathSciNet, so I can calculate my collaboration distance to various friends and acquaintances.

Towards an Erdős–Bacon number of seven

When I noted earlier this year that I was on my way to having an Erdős number of 3, due to either of two upcoming papers, a friend asked if I had an Erdős–Bacon number. My lack of a film career prompted me to answer, "no". I once appeared as an extra in a scene filmed for Lucid Days in Hell, but that scene was cut from the movie. So I didn't see how that helped.

But recently while reading a biography of Jim Henson, another thought came to me: what if I could use TV shows? Henson appeared on a TV show called Afternoon, hosted by Willard Scott and Mac McGarry. I appeared on It's Academic, hosted by Mac McGarry. That's two degrees of separation from Jim Henson! And Willard Scott! But what about Kevin Bacon?

Well, Henson appeared in The Muppet Movie with Austin Pendleton, who appeared in Starting Over with Kevin Bacon. Boom, if you allow TV shows (and you probably shouldn't), I have a Bacon number of four, and an Erdős–Bacon number of seven.

Updated version of "Constructing Carmichael numbers through improved subset-product algorithms"

A new version of "Constructing Carmichael numbers through improved subset-product algorithms" has been posted to the math arXiv.

From the comments: "Table 1 fixed; previously the last 30 digits and number of digits were calculated incorrectly." This now better matches the version that will appear in Math. Comp.

Publication of Constructing Carmichael numbers through improved subset-product algorithms

"Constructing Carmichael numbers through improved subset-product algorithms," co-authored with the late Red Alford, as well as Steven Hayman and Andrew Shallue, has been published on-line by Mathematics of Computation.

It looks like all of the 2013 issues of Math. Comp. are filled up, so this will probably be officially a 2014 paper once it appears in print.

As this is my first co-authored paper to be published, I now have a finite Erdős number, namely 3. (Red co-authored with Carl Pomerance and Andrew Granville, who each have an Erdős number of 1.)

(According to MathSciNet, this reduces Shallue's number from 4 to 3, and is Hayman's first paper. This paper, however, is not indexed by MathSciNet yet, which limits my ability to compute some collaboration distances that interest me, as well as raising the possibility that my co-authors have other un-indexed papers.)

I lost my chance at an Erdős number of 1 by deflecting his questions about what I was working on, but, well, I'm not really into Erdős-style collaborations, and I'm comfortable with my style of research.

SERMON 2013 Talk

I am scheduled to give a talk at the SERMON 2013 conference entitled "Collecting primes with p²-1 1163-smooth, or reduced sets for likely solutions to the $620 problem." It is an update to my 2005 talk.

 Here are the slides.

Repeatedly Appending Digits and Only Finding Composites

I have uploaded slides for my talk at next month's PANTS meeting. The title is "Repeatedly Appending Digits and Only Finding Composites". It is based on joint work with Witold Jarnicki, John Rickert, and Stan Wagon. You can find the paper at Stan's site.

Constructing Carmichael numbers through improved subset-product algorithms

A pre-print of "Constructing Carmichael numbers through improved subset-product algorithms" is now available at the math arXiv. The paper contains research Red Alford and I did. It also contains research by our co-authors, Steven Hayman and Andrew Shallue, who have some impressive constructions and interesting algorithms. I hope to modify the code they wrote to expand the computations of Carmichael numbers with exactly k factors, which was done on a computer that is now outdated.