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.

Google Scholar

I went ahead and claimed my Google Scholar page. I trimmed two articles that I didn't actually write. It claims that I have been cited 85 times, including 33 times in the past 5 years. Perhaps most amusing to me is my work being cited in US Patent #7181017. I'm not actually a fan of patenting algorithms, but I'm glad people are reading my work.

My first Math Reviews byline

Last year, I signed up to be a reviewer for Mathematical Reviews. For those of you not familiar with the publication, it is essentially a database of mathematics articles, with short "reviews" written by other mathematicians. These are not reviews in the ordinary sense of a book or movie review -- the reviewer usually doesn't venture an opinion of the work, and even more rarely expresses a negative one. Rather, the review summarizes the results and attempts to put them into context.

Since 1940, Math Reviews has provided an invaluable service for research mathematicians -- having a good summary of an article is important before one begins the arduous task of tracking down a journal article, and the sometimes more arduous task of reading it. This utility has only increased with the electronic form of the database.

My first review appeared this year, and is of the article "On congruence conditions for primality" in the journal Integers. I think you can access the review here if your institution subscribes, but for various uninteresting reasons, I can't confirm that.


Anyway, I'm particularly proud of my five sentences because I have been a Math Reviews reader for about twenty years, and it is nice to join the ranks of the reviewers. Also, I think I get a discount on my next year's AMS membership.

Letter to the Editor

Stan Wagon wrote a letter to the editor in the April 2011 Mathematics Magazine about Perrin's sequence and Perrin pseudoprimes. You may not be able to access it on-line (I can't), but it reads in part:

An important question is: Is an integer prime if and only if it satisfies the Perrin condition, n divides x_n ? This question was raised by R. Perrin in 1899. A counterexample, now known as a Perrin pseudoprime, was not discovered until 1982: the smallest one is 271441. This is quite remarkable compared to, say, Fermat pseudoprimes with base 2, for which 341 is the smallest example. Recent work by J. Grantham [3] shows that there are infinitely many Perrin pseudoprimes.

It's always gratifying to see one's work referenced, and the letter provides some nice context for my research (better than I did in the paper itself!).
 

John Selfridge, 1927-2010

News has reached me that John Selfridge died at the age of 83 on Halloween.  Selfridge was a pioneer in the area of pseudoprime research; he used what is now known as the strong pseudoprime test to check primality of numbers back in the 1950s.  He was very kind to me when I entered the field as a graduate student.

Along with Pomerance and Wagstaff, he is the originator of what has come to be known as "the $620 problem", the question of whether there is a number which is simultaneously a strong pseudoprime to the base 2 and a Lucas pseudoprime.  If the answer is yes, he promised to pay $500 for the solution, with Wagstaff paying $100 and Pomerance $20.  If the answer is confirmed as no, Selfridge would be on the hook for $20, Wagstaff $100 and Pomerance $500.

When asked why he would volunteer higher amount upon production of a counterexample -- which almost surely exists -- he explained that if someone produced a dense proof claiming to show that none existed, he would rather Pomerance -- with $500 on the line -- have to read through the proof to find an error.  The counterexample, on the other hand, would be easy to verify or dismiss.

Throughout the '90s, and even into this century as his health was failing, I ran into Selfridge at practically every number theory conference I attended.  Having retired, he enjoyed nothing better than traveling around, listening to mathematics talks, and enjoying the social company of mathematicians (with the associated libations).  Having invested well, he decided he would rather the money go to mathematics than to the government through taxes, so he set up the Number Theory Foundation, which helps fund many of the conferences he enjoyed.

Rest in Peace, John Selfridge.  You will be missed.

Le plus grand facteur premier de la fonction de Landau

A preprint appeared on the math arXiv this week entitled, "Le plus grand facteur premier de la fonction de Landau."  I think that translates as, "The largest prime factor of Landau's function."  Landau's function is the maximal order of S_n (the symmetric group on n elements).  The first paper I ever published was The Largest Prime Divisor of the Maximal Order of an Element of S_n.

So this new paper is interesting to me.  My paper is reference 7.  I'm mentioned 3 times in the body of the text.  Here are the mentions, as rendered by Google Translate:

1. This increase was enhanced by Grantham
2. For this we use the method of Grantham
3. In the proof of the theorem of [7], α1 suites. . . , Α9, β1,. . . , Used β9 J. Grantham are very similar to those obtained by Algorithm 1 y = 3329.

I think the last one needs a little work.  I'll have to sit down when I have time to read a 40-page paper in French and figure out what's going on.

But anyway, not only do I have a problem, I also have a method.  Cool.