tag:blogger.com,1999:blog-1364629727248557844.post8604085242321295140..comments2010-03-30T10:15:32.300+11:00Comments on The Mathematical Bricoleur: Just how small is the smallest illegal number?GBhttp://www.blogger.com/profile/01100913350264083875noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-1364629727248557844.post-51881286209074426942009-01-07T17:57:00.000+11:002009-01-07T17:57:00.000+11:00Yes, Peter, I am.Yes, Peter, I am.GBhttps://www.blogger.com/profile/01100913350264083875noreply@blogger.comtag:blogger.com,1999:blog-1364629727248557844.post-38050971074369962042009-01-07T17:32:00.000+11:002009-01-07T17:32:00.000+11:00You didn't put any profile information, but I surm...You didn't put any profile information, but I surmise from the time stamp that you are in Eastern Australia. Am I right?Peterhttps://www.blogger.com/profile/04875668909485036128noreply@blogger.comtag:blogger.com,1999:blog-1364629727248557844.post-44061254017339156612009-01-07T17:23:00.000+11:002009-01-07T17:23:00.000+11:00Some of what is being said is a little above my he...Some of what is being said is a little above my head, but it's good for me. Amazing coincidence: Just today I told my intro to algebra students that I had read all finite sequences of digits appear in pi. We had fun looking up a fake phone number on a pi website. They were amazed to hear that this all implies that the complete works of Shakespeare (using a=1, b=2,...) must appear.Peterhttps://www.blogger.com/profile/04875668909485036128noreply@blogger.comtag:blogger.com,1999:blog-1364629727248557844.post-35130832740903845302009-01-06T10:33:00.000+11:002009-01-06T10:33:00.000+11:00Actually, your proof draws a somewhat incorrect co...Actually, your proof draws a somewhat incorrect conclusion. The conclusion *isn't* that all numbers are illegal, merely that illegal numbers are incomputable. The proof is a direct analogue of Berry's paradox, or the proof of the incomputability of Kolmogorov Complexity. (KC)<BR/><BR/>This of course leads to the following definition of illegal number:<BR/><BR/>An illegal number is a number of sufficient length which encodes proprietary information and whose KC to size ratio is sufficiently small. In other words, compressible numbers which "mean" something.<BR/><BR/>Even so, one could always take a One-Time Pad as an encryption key, either on the industry side, or the pirating side - thereby essentially transferring random bits; then the original number must be under copyright, and all of its encryptions under the OTP-scheme. Therefore, all sufficiently large numbers are illegal.<BR/><BR/>I suppose the way to reason around this is as follows: A number is associatively illegal if, under a known and sufficiently widespread, *bijective* compression and/or encryption function, it is the image of an illegal number.<BR/><BR/>It follows that all numbers are either illegal or associatively illegal; however to prove wrongdoing, one would have to prove that the person used an associatively illegal number in conjunction with a specific compressor/encryptor to represent a specific illegal number; which I think is what we want.rhapsody407https://www.blogger.com/profile/16782680590749008969noreply@blogger.comtag:blogger.com,1999:blog-1364629727248557844.post-37126310332909295202009-01-06T08:51:00.000+11:002009-01-06T08:51:00.000+11:00I think the induction argument applies to the algo...I think the induction argument applies to the algorithm as well. That is, you'd have to post 0 somewhere and then also post an algorithm for reaching the smallest illegal number from 0. So far you've only posted "keep adding one until the RIAA knocks down your door". Could take awhile.Enginerdhttps://www.blogger.com/profile/07211929551692433064noreply@blogger.com