Carmichael Numbers with Exactly k Prime Factors

One of my projects for this year is to finish a paper that Red Alford and I started where we show the existence of Carmichael numbers with exactly k prime factors for a wide range of k.  Red passed away 7 years ago, so I think it's about time I get this paper out of the door.

One of my problems has been with the complexity of the code needed to finish the computation.  Recently I've been teaching myself Python, which looks like it can cut through some of the complexity issues for the less computationally-intensive parts of the computation.

Anyway, I recently received a request for information about the technique Red and I used.  I realized that I never put any of my talks about this technique on-line, so I uploaded a talk I gave in 2003 at the Hugh Williams conference in Banff.

Online publication

My article on the infinitude of Perrin pseudoprimes has now been published on-line.  Paper publication to follow. 

"Please note access to the full text of this article will depend on your personal or institutional entitlements."

Now, with keywords!

The Journal of Number theory has asked me to add keywords to the paper.  I also added them to my own copy, the updated version of which is available on Google Docs.

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Yet another new version of the paper

I got the proofs back from the Journal of Number Theory, and they made a few minor changes (mostly capitalization and such).  I'm having trouble ftp'ing to pseudoprime.com, so you can find the latest versions here on Google Docs.

Even newer version of paper!

The referee had 9 additional suggestions after receiving my previous revision. I incorporated 8 of them, and the paper has now officially been accepted by the Journal of Number Theory. The latest version is here.

Elliptic Pseudoprimes

Mathematics of Computation recently published an article by Siguna Müller entitled, "On the existence and non-existence of elliptic pseudoprimes". She refers to two of my papers (references [7] and [8]).




I have not looked much at elliptic pseudoprimes. In particular, in my 2001 paper, "Frobenius Pseudoprimes", I say,

This paper does not pretend to be an exhaustive treatment of all notions of pseudoprimality. For example, nothing is said about elliptic pseudoprimes [8].

Maybe I should look at them some more.

Newer version of paper!

I have incorporated the referee's 44 suggestions and made other changes. The newest version of the paper is available here. The editor is sending it back to the referee. Hopefully this merry-go-round will stop soon!

Grantham's Problem

While reviewing the referee's 44 (sigh) suggested changes to my paper, I came across an article published last November entitled "Inefficacious Conditions of the Frobenius Primality Test and Grantham's Problem".

I have a problem named after me!

So the next time someone asks me, "What's your problem?" I can say, "Are there any composite numbers n ≡ ±2 (mod 5) such that x^(n+1) ≡ 5 (mod(n, x^2 + 5x + 5))?"

New Version of Paper

Well, this time I didn't take 6 or 7 years. After slightly more than two years, I've revised There Are Infinitely Many Perrin Pseudoprimes again. I used a more modern version of the journal's style guide, and I incorporated some comments an editor suggested. Now hopefully it'll go off to another editor (don't ask) and a referee!

Things Have Changed

Since I last submitted that paper.

1. The journal that had been interested in publishing it no longer publishes things on that subject.
   
2. A paper can be turned down in 2 days.
   

Still, the "rejection" was very encouraging, saying

I would urge the author to submit his paper to a top-end number theory journal...

So I will, but I have to do some further (minor) reformatting.

New Version of "There Are Infinitely Many Perrin Pseudoprimes"

I have reformatted There Are Infinitely Many Perrin Pseudoprimes (sometimes known as "There Are Infinitely Many Frobenius Pseudoprimes"). It now uses a more modern version of the TeX typesetting package. I also re-submitted it for publication after only 6 or 7 years. I think it holds up, though!

Corrected SERMON slides

I have posted a corrected version of my slides from SERMON.

SERMON 2005 Talk

I am giving a talk at the SERMON 2005 conference entitled, "Collecting primes with p²-1 827-smooth, or reduced sets for likely solutions to the $620 problem."

Here are the slides.

Welcome to the Blog

I have create this new weblog to post updates on my mathematical research. It will not necessarily be updated frequently, but will contain links to talks, papers, and other items of interest.