tag:blogger.com,1999:blog-264226589944705290.post7119370900672169780..comments2023-11-05T03:45:25.001-08:00Comments on God Plays Dice: Numbers that "look prime"Michael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-264226589944705290.post-62963029886045734632007-11-25T13:38:00.000-08:002007-11-25T13:38:00.000-08:00John, a nice reference for these 'repunits' as the...John, a nice reference for these 'repunits' as they are called is the <A HREF="http://mathworld.wolfram.com/Repunit.html" REL="nofollow"><BR/>MathWorld article</A>.<BR/>A necessary condition for the nth repunit, a string of n 1's, to be prime is that n is prime (otherwise it's easily factorable) but this is not a sufficient condition. Lots of research about these kinds of numbers.Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-38861890960998941122007-11-25T09:50:00.000-08:002007-11-25T09:50:00.000-08:00Yeah, I have to agree with jack. Multiples of 11 ...Yeah, I have to agree with jack. Multiples of 11 do have an easy test, even in many digits. A number is divisible by 11 if and only if (I think) the sum of the odd position digits is equal to the sum of the even position digits. It's not quite as nice as 2,3,5 but it's workable.janotharhttps://www.blogger.com/profile/00118523275073693196noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-35707718534732459282007-11-25T08:56:00.000-08:002007-11-25T08:56:00.000-08:00Which ones are easy to eliminate?Any composite len...<I>Which ones are easy to eliminate?</I><BR/><BR/>Any composite length goes right away. Three is also out by inspection. The rest are tough, but the 19th and 23rd in the sequence seem to be prime, so they're not all composite.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-73730850317171978742007-11-25T05:41:00.000-08:002007-11-25T05:41:00.000-08:00One of my favorite patterns is 1,11,111,1111,11111...One of my favorite patterns is 1,11,111,1111,11111,... Are any of these prime after 11? Which ones are easy to eliminate? A related sequence would be 101,1001,10001,...<BR/>Lots of interesting connections to algebra here (which is of course part of pure number theory).Dave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-86285022277101242102007-11-24T18:31:00.000-08:002007-11-24T18:31:00.000-08:00I'll follow Grothendieck. 57 is prime!I'll follow Grothendieck. 57 <I>is</I> prime!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-36129144366681713702007-11-24T14:41:00.000-08:002007-11-24T14:41:00.000-08:00Also, 253 doesn't look prime to me, because the mi...Also, 253 doesn't look prime to me, because the middle digit is the sum of the other two, which makes it a multiple of 11.<BR/><BR/>Jack<BR/>(1979 Texas Mental Arithmetic Champion!)I. J. Kennedyhttps://www.blogger.com/profile/04805435564360543720noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-7917775074499214372007-11-24T10:55:00.000-08:002007-11-24T10:55:00.000-08:00I reread my comment, and I noticed it could be mis...I reread my comment, and I noticed it could be misunderstood. Of course 143 is not a prime number; I wanted to say that - at least for me, like 217 and 287 for Isabel - when I see it I immediately know it is not a prime number..mau.https://www.blogger.com/profile/09641196427325175260noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-33005081503043239562007-11-24T06:41:00.000-08:002007-11-24T06:41:00.000-08:00Isabel,You're absolutely right - this is harder th...Isabel,<BR/>You're absolutely right - this is harder than it looks! Obviously, most of our middle schoolers and high schoolers haven't worked enough with the sequence of primes up to 150 or so to be fluent. One of the purposes or benefits of activities like this one is to provide enough practice with primes <BR/>so that students who ordinarily freeze up under pressure or don't pay much attention will actually improve over time! Their confidence, accuracy and speed become progressively better so that more students remain standing. Initially, your best students will generally survive the first couple of rounds and prevail, not only because they have more fluency with primes and better retention but, most importantly, because of their ability to focus and concentrate which separate them from the rest. As an educator I recognize that some students are naturally more capable in that area but I do not then assume that others in the class cannot improve their performance via many such activities. We (not you!) often overlook that activities such as mathematics, playing chess, etc., develop our powers of concentration as well as our reasoning ability.<BR/><BR/>By the way, I view myself as both a math educator and a mathematician, but my blog is most often classified as a 'math ed blog', I guess because I write activities for the classroom teacher. Oh well... <BR/><BR/>I realize that your focus was more on numbers that appear to be prime but aren't but I did want to mention this issue of cognitive development that is also important to me. <BR/><BR/>Isabel, thanks again for linking your many readers to my blog.<BR/><BR/>Dave Marain<BR/>MathNotationsDave Marainhttps://www.blogger.com/profile/13321770881353644307noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-64296930257397979912007-11-24T04:09:00.000-08:002007-11-24T04:09:00.000-08:00143 is ostensibly one less of 144 which is a squar...143 is ostensibly one less of 144 which is a square, so I would not say it could be a prime :-).mau.https://www.blogger.com/profile/09641196427325175260noreply@blogger.com