At 3:19 AM 1/5/96, jim bell wrote:
But BTW, isn't it interesting, that news item from a few weeks ago, on an algorithm for determining individual bits in Pi, regardless of whether you've calculated all the previous ones. Only problem is, it only works in hexadecimal (and, obviously, binary, etc, not decimal. ^^^^^^^^^^^
??? I didn't see this result you mention, but it surprises me. The part about how it works in some bases, but not in decimal. I assume it really works only in binary, and hexadecimal follows, not the other way around. The "hand-waving" (motivational/informal) explanation for why I am surprised is that "Nature doesn't care about bipeds with 10 digits vs. bipeds, or whatever, with 2 digits or 16 digits." That is, results applicable in base 16, hexadecimal, should be easily applicable in base 10. Sorry, but it is quite possible for this to be the case. (I don't know for sure whether this is one of them or not, though, having not seen the result myself.) But assume for the moment that the formula, or algorithm, or whatever it is, really does tell you exactly the value of a contiguous chunk of "bits", real honest-to-god binary digits. You cannot translate these to a decimal representation without knowing all of the bits leading up to them. For example, you know the last four bits of an eight bit string: XXXX0011 In Hex the last digit is 3. But what is the last digit in decimal? If the 'X's are all 0, it is 3, but if the last X is a 1 (making the number 00010011 = 19), it is not 3 but 9. If only the first X is a one, it is 1. There are plenty of places in information theory where a log base 2 shows up, so I don't doubt that there might be an algorithm for determining a particular "bit" of Pi. But just to prove I have a more concrete example, suppose you have an encrypted bank transfer, with the numbers expressed in binary. Further suppose you know it is encrypted with a one-time-pad (just to be contraversial) where you know a particular n-bit chunk of the pad. Given this you can recover the corresponding n-bit chunk of the amount, but unless this spans the entire number you can't express this unambiguously in decimal digits. This is a simple consequence of the fact that log(2) and log(10) are not integer multiples of each other (you know what I mean). The same goes the other way, of course. Given a string of decimal digits extracted from the middle of a number, I can't unambiguously decide what string of bits these would become without knowing the rest of the number. The result is fascinating, assuming it is real. Greg. Greg Rose INTERNET: greg_rose@sydney.sterling.com Sterling Software VOICE: +61-2-9975 4777 FAX: +61-2-9975 2921 28 Rodborough Rd. http://www.sydney.sterling.com:8080/~ggr/ French's Forest 35 0A 79 7D 5E 21 8D 47 E3 53 75 66 AC FB D9 45 NSW 2086 Australia. co-mod sci.crypt.research, USENIX Director.