tag:blogger.com,1999:blog-6642011.post110998067249458011..comments2020-10-11T06:04:29.722-04:00Comments on Philosophy, et cetera: Constructing the Largest PrimeRichard Y Chappellhttp://www.blogger.com/profile/16725218276285291235noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6642011.post-1110225709620779782005-03-07T15:01:00.000-05:002005-03-07T15:01:00.000-05:00Hi Brian - nice to hear from you! I was hoping th...Hi Brian - nice to hear from you! I was hoping that the messy parenthetical section of my final paragraph might complete the construction (since it either determines m to be prime, or else finds its smallest factor, which would itself be prime), but I wasn't very clear. I'm not even entirely sure it works, since (as I'm sure <I>is</I> clear) I know very little about all this stuff!Richard Y Chappellhttps://www.blogger.com/profile/16725218276285291235noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1110202027031111352005-03-07T08:27:00.000-05:002005-03-07T08:27:00.000-05:00Obviously what Richard said is correct, so intuiti...Obviously what Richard said is correct, so intuitionists needn't worry about this, but there is still a misleading aspect to the post. What matters is that the prime be constructable, but it isn't clear that what you've given is yet a construction. A proof that not both of a and b is not-F is not a construction of an F. So a proof that not all of the numbers up to n!+1 are composite is not yet a proof that one of them is prime.<br /><br />As it turns out in this case, it is a finite procedure to work out whether a particular number is prime, so once you've proven that not all the numbers in a finite set are composite, it is provable that one of them is prime. But you shouldn't be too content to rest with general conclusions like not all of the numbers through n!+1 are composite.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1109985333912439532005-03-04T20:15:00.000-05:002005-03-04T20:15:00.000-05:00It is intuitionistically provable. It is even fin...It is intuitionistically provable. It is even finitarily provable (see Hilbert's "On the infinite"). Of course, if you're an ultrafinitist, you'd believe it false.Legacy Userhttps://www.blogger.com/profile/13925247685067322932noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1109982880473138242005-03-04T19:34:00.000-05:002005-03-04T19:34:00.000-05:00Yes, that was a bit sloppy of me! I'll update my ...Yes, that was a bit sloppy of me! I'll update my post to make that clearer... thanks for the pointer.Richard Y Chappellhttps://www.blogger.com/profile/16725218276285291235noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-1109982711001008982005-03-04T19:31:00.000-05:002005-03-04T19:31:00.000-05:00Be careful! Some might think you are saying that ...Be careful! Some might think you are saying that <I>p</I>! + 1 is always prime for prime <I>p</I>, but this is not the case; for example, 5! + 1 = 120 + 1 = 121 = 11^2. However, it is true that the prime factors of <I>p</I>! + 1 are all greater than <I>p</I>, so one can get a larger prime by factoring that. <br /><br />I'm by no means knowledgeable about the philosophy of mathematics, though. Good luck finding answers to your questions!Kevinhttps://www.blogger.com/profile/16536296799947870206noreply@blogger.com