tag:blogger.com,1999:blog-121788532019-04-13T09:30:38.495-04:00Pseudoprimes and Other ResearchItems related to Jon Grantham's mathematical research.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.comBlogger40125tag:blogger.com,1999:blog-12178853.post-60638931893622056252019-04-13T09:30:00.000-04:002019-04-13T09:30:38.486-04:00Talk on Brazilian Primes at SERMON 2019Here are <a href="http://www.pseudoprime.com/sermon19a.pdf">the slides of the talk</a> I am giving today at the <a href="https://mathstats.uncg.edu/number-theory/conferences/sermon-2019/">SERMON 2019</a> 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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-69757912769271148262019-03-20T15:33:00.001-04:002019-03-20T15:33:03.369-04:00Moving to the arXivWhen 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.<br /><br />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.<br /><br />So you can now see all of my reprints (and <a href="http://math.pseudoprime.com/2019/03/brazilian-primes-which-are-also-sophie.html">preprint!</a>) at <a href="https://arxiv.org/a/grantham_j_1.html">this arXiv link</a>.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-49567829656312708112019-03-13T12:05:00.000-04:002019-03-13T12:05:02.905-04:00Brazilian Primes Which Are Also Sophie Germain PrimesA preprint of "Brazilian Primes Which Are Also Sophie Germain Primes" is <a href="https://arxiv.org/abs/1903.04577">now available on the math arXiv</a>. In it, Hester Graves and I disprove a conjecture from <a href="https://oeis.org/A125134/a125134.pdf">Schott's 2010 paper</a> on Brazilian primes, namely that no Brazilian primes are also Sophie Germain primes.<br /><br />What does this mean? A Brazilian prime is a prime number of the form 1+b+b<sup>2</sup>+...+b<sup>k-1</sup>, 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".<br /><br />A Sophie Germain prime is a prime p such that 2*p+1 is also prime.<br /><br />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.<br /><br />In the paper, we computed all Brazilian Sophie Germain primes up to 10<sup>44</sup>. 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 10<sup>52</sup>.<br /><br />We have actually computed all Brazilian Sophie Germain primes up to 10<sup>46</sup> (there are 104,890,302 of them) and 10<sup>48</sup> (we haven't counted them up yet). A later version of the preprint will reflect that.<br /><br />Submission of the sequence of Brazilian Sophie Germain primes is in progress. A later version of the preprint will also reflect that.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-24658932563094382292019-02-23T11:47:00.004-05:002019-02-23T11:47:57.398-05:00Primes Which Are Values of Cyclotomic PolynomialsIn 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 <a href="https://tigerweb.towson.edu/nmcnew/mason3/">MASON III conference</a> at James Madison University. <a href="http://www.pseudoprime.com/mason19.pdf">Here are the slides.</a>Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-92185205506111793282019-01-19T13:18:00.002-05:002019-01-19T13:18:29.463-05:00Cyclotomic Goldbach<a href="https://3.bp.blogspot.com/-t2uqP1Wa4NI/XENpx93V0cI/AAAAAAACRUM/AJIs91Zg860RmQL1o9dF4tWeg43mQpJawCKgBGAs/s1600/JPEG_20181217_090721_8867237943679701403.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="1600" height="240" src="https://3.bp.blogspot.com/-t2uqP1Wa4NI/XENpx93V0cI/AAAAAAACRUM/AJIs91Zg860RmQL1o9dF4tWeg43mQpJawCKgBGAs/s320/JPEG_20181217_090721_8867237943679701403.jpg" width="320" /></a>Last month, at the <a href="https://westcoastnumbertheory.org/">West Coast Number Theory conference</a> 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. <a href="http://www.pseudoprime.com/wcnt18.pdf">Here are the slides</a>.<br /><br /><br />Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-70626505137974780802018-04-06T18:42:00.003-04:002018-04-06T18:42:40.103-04:00Talking Again about Another Conjecture of Goldbach<a href="http://www.pseudoprime.com/mason18.pdf">Here are the slides</a> for a talk I am giving tomorrow at the <a href="https://tigerweb.towson.edu/nmcnew/mason2/">MASON conference</a> in Towson, MD. They are almost identical to the slides for the <a href="http://math.pseudoprime.com/2018/03/another-conjecture-of-goldbach.html">talk I gave last month at the SERMON conference.</a><br /><br />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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-9960900296666110022018-03-08T14:22:00.004-05:002018-03-08T14:22:35.785-05:00Another Conjecture of Goldbach<a href="http://www.pseudoprime.com/sermon18.pdf">Here are the slides</a> for a talk I am giving this weekend at the <a href="http://faculty.etsu.edu/keatonr/SERMON.aspx">SERMON conference</a> in Johnson City, TN. The basic question addressed is as follows.<br /><br />Look at all numbers <i>a</i> such that <i>a<sup>2</sup>+1</i> 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 <i>a</i> up to 2*10<sup>14</sup> and explore how easy it is to find these sums. This is joint work with Hester Graves.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-53482860912106584392016-10-29T12:08:00.002-04:002016-10-29T12:08:54.611-04:00Parallel Computation of Primes of the Form x2+1Today I gave a talk on Parallel Computation of Primes of the Form x<sup>2</sup>+1 at Towson University, at the <a href="https://tigerweb.towson.edu/nmcnew/mason/">first MASON conference</a>. <br /><a href="https://drive.google.com/file/d/0B_rx1Wv-LXdnU1VMbm1UcXdQUEU/view?usp=sharing">You can see my slides here</a>.<br /><br />I computed all such primes up to 6.25x10<sup>28</sup>. Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-64372286408160643832015-08-18T19:08:00.003-04:002015-08-18T19:08:26.863-04:00Talk at Pomerance Conference<div class="separator" style="clear: both; text-align: center;"><a href="http://alpha.math.uga.edu/~cp70/CP70/Home_files/shapeimage_2.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://alpha.math.uga.edu/~cp70/CP70/Home_files/shapeimage_2.png" height="169" width="320" /></a></div>Slides for my talk from <a href="http://alpha.math.uga.edu/~cp70/CP70/Home.html">June's Pomerance conference</a> are now <a href="http://alpha.math.uga.edu/~cp70/CP70/Schedule_files/grantham.pdf">available at the conference website</a>.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-71458598431448549542014-05-13T14:15:00.002-04:002014-05-13T14:15:16.621-04:00All My ReprintsInspired by the availability of a reprint for my latest publication, I have updated my <a href="http://www.pseudoprime.com/jgpapers.html">list of reprints</a>. 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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-85871992840428139822014-05-13T13:20:00.004-04:002014-05-13T13:20:51.673-04:00Full Text of "Repeatedly Appending Any Digit to Generate Composite Numbers"The <i>American Mathematical Monthly</i> 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 <a href="http://www.arxiv.org/" target="_blank">www.arXiv.org</a> and your personal website if you so choose." Very reasonable!<br /><br />It is available at <a href="http://www.pseudoprime.com/amer.math.monthly.121.05.416-wagon.pdf">http://www.pseudoprime.com/amer.math.monthly.121.05.416-wagon.pdf</a>.<br /><br />So you can read it even if you don't otherwise have access to the <i>Monthly</i>, which Wikipedia <a href="https://en.wikipedia.org/wiki/American_Mathematical_Monthly">tells me</a> is the "most widely read mathematics journal in the world." Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-10119854139264414632014-04-18T13:13:00.000-04:002014-04-18T13:13:15.316-04:00Publication of "Repeatedly Appending Any Digit to Generate Composite Numbers"<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-jwe_hCZfrAE/U1Fct-uGCkI/AAAAAAABdF8/w10k7mqu1oo/s1600/Screenshot-13.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-jwe_hCZfrAE/U1Fct-uGCkI/AAAAAAABdF8/w10k7mqu1oo/s1600/Screenshot-13.png" height="216" width="320" /></a></div>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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-62843010381450679172013-12-18T16:02:00.001-05:002013-12-18T16:02:16.867-05:00Print 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 <i>Mathematics of Computation</i>.<br /><br />That means if you want to cite it, you can now cite it as:<br /> <span class="title">Constructing Carmichael numbers through improved subset-product algorithms.</span> <em>Math. Comp.</em> 83 (2014), no. 286, 899-915.<br /><br />I'm still waiting for it to appear in MathSciNet, so I can calculate my collaboration distance to various friends and acquaintances. Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-8212609814135997012013-11-04T10:31:00.000-05:002013-11-04T10:31:18.624-05:00Towards an Erdős–Bacon number of seven<a href="http://commons.wikimedia.org/wiki/File%3AErdos_head_budapest_fall_1992.jpg" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" title="By Kmhkmh (Own work) [GFDL (http://www.gnu.org/copyleft/fdl.html) or CC-BY-3.0 (http://creativecommons.org/licenses/by/3.0)], via Wikimedia Commons"><img alt="Erdos head budapest fall 1992" src="//upload.wikimedia.org/wikipedia/commons/f/f1/Erdos_head_budapest_fall_1992.jpg" width="128" /></a><a href="http://commons.wikimedia.org/wiki/File%3AKevin_Bacon_(cropped).jpg" title="By SAGIndie from Hollywood, USA (Flickr) [CC-BY-2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons"><img alt="Kevin Bacon (cropped)" src="//upload.wikimedia.org/wikipedia/commons/1/1a/Kevin_Bacon_%28cropped%29.jpg" width="128" /></a> When <a href="http://math.pseudoprime.com/2013/07/publication-of-constructing-carmichael.html">I noted</a> earlier this year that I was on my way to having an <a href="https://en.wikipedia.org/wiki/Erd%C5%91s_number">Erdős number</a> 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 <a href="http://www.imdb.com/title/tt0270485/">Lucid Days in Hell</a>, but that scene was cut from the movie. So I didn't see how that helped.<br /><br />But recently while reading <a href="http://www.amazon.com/gp/product/B00C4BA5A8/ref=as_li_ss_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=B00C4BA5A8&linkCode=as2&tag=twinpaniccom-20">a biography of Jim Henson</a><img alt="" border="0" height="1" src="http://ir-na.amazon-adsystem.com/e/ir?t=twinpaniccom-20&l=as2&o=1&a=B00C4BA5A8" style="border: none !important; margin: 0px !important;" width="1" />, another thought came to me: what if I could use TV shows? Henson appeared on a TV show called <a href="http://muppet.wikia.com/wiki/Afternoon_with_Inga">Afternoon</a>, 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?<br /><br />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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-27355489472665748622013-09-08T15:21:00.000-04:002013-09-08T15:21:08.689-04:00Updated version of "Constructing Carmichael numbers through improved subset-product algorithms"A new version of "Constructing Carmichael numbers through improved subset-product algorithms" <a href="http://arxiv.org/abs/1203.6664">has been posted</a> to the math arXiv.<br /><br />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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-13313710886423293102013-07-16T17:45:00.002-04:002013-07-16T17:45:18.753-04:00Publication 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, <a href="http://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2013-02737-8/">has been published on-line by <i>Mathematics of Computation</i></a>.<br /><br />It looks like all of the 2013 issues of <i>Math. Comp.</i> are filled up, so this will probably be officially a 2014 paper once it appears in print.<br /><br />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.)<br /><br />(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.)<br /><br />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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-5285007441118998302013-04-12T23:52:00.000-04:002013-04-12T23:52:00.257-04:00SERMON 2013 TalkI am scheduled to give a talk at the SERMON 2013 conference entitled "Collecting primes with <tt>p<sup>2</sup>-1</tt> 1163-smooth, or reduced sets for likely solutions to the $620 problem." It is an update to <a href="http://math.pseudoprime.com/2005/04/sermon-2005-talk.html">my 2005 talk</a>.<br /><br /><a href="http://www.pseudoprime.com/pseudo/sermon13.pdf"> Here are the slides.</a>Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-28487671162384934852012-08-20T09:21:00.000-04:002012-08-20T09:21:05.197-04:00Repeatedly Appending Digits and Only Finding CompositesI have <a href="http://www.pseudoprime.com/pants12.pdf">uploaded slides</a> for my talk at <a href="http://www.wfu.edu/%7Erouseja/pantsxviii/">next month's PANTS meeting</a>. 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 <a href="http://stanwagon.com/public/GranthamRickertJarnickiWagonFinalJuly2012.pdf">at Stan's site</a>.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-73585548176320653122012-04-02T11:24:00.004-04:002012-04-02T11:24:34.405-04:00Constructing Carmichael numbers through improved subset-product algorithmsA pre-print of "<a href="http://arxiv.org/abs/1203.6664">Constructing Carmichael numbers through improved subset-product algorithms</a>" 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.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-18196059175515382422011-12-07T01:30:00.001-05:002011-12-07T01:33:42.662-05:00Google ScholarI went ahead and claimed my <a href="http://scholar.google.com/citations?user=GfmpnGYAAAAJ&hl=en">Google Scholar page</a>. 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 <a href="http://www.google.com/patents?hl=en&lr=&vid=USPAT7181017&id=JPx-AAAAEBAJ&oi=fnd&printsec=abstract#v=onepage&q&f=false">US Patent #7181017</a>. I'm not actually a fan of patenting algorithms, but I'm glad people are reading my work.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-53977915457703731002011-05-03T13:03:00.000-04:002011-05-03T13:03:39.522-04:00My first Math Reviews byline<a href="http://1.bp.blogspot.com/-bGBkWIbUZ5s/TcA1OJIEg7I/AAAAAAAARgY/mN5EtvveARA/s1600/Screenshot-1.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="236" src="http://1.bp.blogspot.com/-bGBkWIbUZ5s/TcA1OJIEg7I/AAAAAAAARgY/mN5EtvveARA/s400/Screenshot-1.png" width="400" /></a>Last year, I signed up to be a reviewer for <a href="http://www.ams.org/mr-database">Mathematical Reviews</a>. 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.<br /><br />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.<br /><br />My first review appeared this year, and is of the article "<span class="title"><span class="searchHighlight">On congruence conditions for primality" in the journal <a href="http://www.integers-ejcnt.org/">Integers</a>. I think you can access the review <a href="http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=On+congruence+conditions+for+primality&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&s7=&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&review_format=html">here</a> if your institution subscribes, but for various uninteresting reasons, I can't confirm that.</span></span><br /><span class="title"><span class="searchHighlight"><br /></span></span><br /><span class="title"><span class="searchHighlight">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 <a href="http://www.ams.org/">AMS</a> membership.</span></span>Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-27227974409155612712011-03-22T12:49:00.000-04:002011-03-22T12:49:46.127-04:00Letter to the Editor<a href="http://stanwagon.com/">Stan Wagon</a> wrote a <a href="http://www.jstor.org/stable/10.4169/math.mag.84.2.119">letter to the editor</a> 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:<br /><blockquote>An important question is: Is an integer prime if and only if it satisﬁes the Perrin condition, n divides x<sub>n</sub> ? This question was raised by R. Perrin in 1899. A counterexample, now known as a <i>Perrin pseudoprime</i>, 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 [<b>3</b>] shows that there are inﬁnitely many Perrin pseudoprimes.</blockquote>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!). <br /> Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-28129378071239711422010-11-15T14:44:00.000-05:002010-11-15T14:44:56.155-05:00John Selfridge, 1927-2010<div class="separator" style="clear: both; text-align: center;"><a href="http://www.ams.org/images/Selfridge.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://www.ams.org/images/Selfridge.jpg" /></a></div><a href="http://www.legacy.com/obituaries/daily-chronicle/obituary.aspx?n=john-selfridge&pid=146547346&fhid=6514">News has reached me</a> 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.<br /><br />Along with Pomerance and Wagstaff, he is the originator of what has come to be known as <a href="http://math.pseudoprime.com/2005_04_01_archive.html">"the $620 problem"</a>, 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.<br /><br />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.<br /><br />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.<br /><br />Rest in Peace, John Selfridge. You will be missed.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-770247857673752452010-09-16T15:52:00.000-04:002010-09-16T15:52:40.772-04:00Le plus grand facteur premier de la fonction de LandauA preprint appeared on the math arXiv this week entitled, "<a href="http://arxiv.org/abs/1009.2944">Le plus grand facteur premier de la fonction de Landau</a>." I think that translates as, "The largest prime factor of Landau's function." Landau's function is the maximal order of S<sub>n</sub> (the symmetric group on n elements). The first paper I ever published was <a href="http://www.pseudoprime.com/maxord.html">The Largest Prime Divisor of the Maximal Order of an Element of S<sub>n</sub></a>.<br /><br />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:<br /><ol><li><span class="short_text" id="result_box"><span title="">This increase was enhanced by Grantham</span></span></li><li><span class="short_text" id="result_box"><span title="">For this we use</span><span title=""> the method of Grantham</span></span><span class="short_text" id="result_box"><span title=""> </span></span></li><li><span class="" id="result_box"><span title="">In the proof of the theorem of [7], α1 suites. </span><span title="">. </span><span title="">. </span><span title="">, Α9, β1,. </span><span title="">. </span><span title="">. </span><span title="">, Used β9</span><span style="background-color: white;" title=""> J. </span><span style="background-color: white;" title="">Grantham are very similar to those obtained by Algorithm 1</span><span style="background-color: white;" title=""> y = 3329.</span></span></li></ol><br />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.<br /><br />But anyway, not only do I have <a href="http://www.pseudoprime.com/pseudo/2009/09/granthams-problem.html">a problem</a>, I also have a method. Cool.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0tag:blogger.com,1999:blog-12178853.post-45906569978821161522010-08-24T14:46:00.001-04:002012-10-26T17:36:17.777-04:00Carmichael Numbers with Exactly k Prime FactorsOne 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.<br /><br />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.<br /><br />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 <a href="http://www.pseudoprime.com/pseudo/williams03.pdf">uploaded</a> a talk I gave in 2003 at <a href="http://www.fields.utoronto.ca/programs/scientific/02-03/numtheory/">the Hugh Williams conference in Banff</a>.Jon Granthamhttp://www.blogger.com/profile/10730444988594582302noreply@blogger.com0