source file: mills2.txt Date: Thu, 21 Dec 1995 02:31:24 -0800 Subject: Re: Transcendental numbers From: jpff@maths.bath.ac.uk Message written at 20 Dec 1995 21:16:51 +0000 In-reply-to: <0099B1E361BED53A.4D16@ezh.nl> (message from Manuel Op de Coul on Tue, 19 Dec 1995 12:48:28 -0800) Manuel has covered most of the points, but I though I would add that my friend Yasamasa Kanada holds the record for calculating teh digits of pi, and teh apparently random nature extends all the way to the 4 billion digits he has calculated and verified. (actually 4,294,960,000 digits are certainly correct) Frequency distribution for pi-3 up to 3,000,000,000 decimal places: '0' : 299999143; '1' : 299995932; '2' : 299989126; '3' : 299992290 '4' : 300002257; '5' : 299979016; '6' : 300025447; '7' : 299975510 '8' : 300016550; '9' : 300024729; Chi square = 9.24 Frequency distribution for 1/pi up to 3,000,000,000 decimal places: '0' : 300009029; '1' : 300002431; '2' : 299992729; '3' : 299998519 '4' : 299970883; '5' : 299980175; '6' : 300010123; '7' : 300025696 '8' : 300008447; '9' : 300001968; Chi square = 7.40 Yasamasa tells me he has gne way beyond this record but the details are embargoed until January, but the distributions seem the same. What is always odd is that the other well-known transcendental number e is very un-random. Strange. My father always used to ask me to give him a transcendental which was not pi, e or multiples; after all there are very many of them. I agree that e+pi is not decided -- there is a theorem by W.S.Brown which is significant for computer algebra based on a conjecture which has e+pi being transcendental as a specal case, and I have not heard that it was decided. My PhD depends in part on Gelfond-Schneider theorem, and I used to search this stuff. Oh well, enough memory lane for tonight. ==John ff Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Thu, 21 Dec 1995 14:51 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id FAA07672; Thu, 21 Dec 1995 05:51:03 -0800 Date: Thu, 21 Dec 1995 05:51:03 -0800 Message-Id: <0099B33C0E28C9B9.5145@ezh.nl> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu