Hello, I agree with Robert Munafo, cracking real numbers is hard, as far as I know there are 4-5 ways to do that. 1) Use my inverter with all the imperfections and these huge tables. Some findings have been made based on the version in Vancouver. The current one has 5.213 billion entries, some is available here http://pictor.math.uqam.ca/~plouffe/pi/ip/ in raw format, beware there are 9000 files totalling 657 gigabytes of data. 2) Use the Maple routine called identify(); well it gives neat examples in some cases but misses many others, with the option 'all' it can find things sometimes. 3) Use the RIES program, very fast yes, can find genuine approximations in a jiffy, better than any other methods but if we follow Steven Finch approach to the problem there are many improvements we could make by using named constants, there are hundreds of them like the Madelung constants, the Riemann zeros, the parking constant, the sofa constant, the Feigenbaum constants, etc. 4) Use a version of a generalized expansion program, it can find some continued fractions, some exotic expansions, but falls short on most real numbers and produces too much usable data. I made many versions of this idea and came out with more than 500 ways to expand a real number into a sequence. Some findings were made using this idea. It could be generalized even more but a good questions is , yes but in what direction ?? 5) The LLL-PSLQ algorithm, excellent for narrow cases but cannot find any of the neat examples suggested here recently like this approximation within 13 digits of a factorial expression found by the RIES program. In some cases like finding an algebraic expression the LLL-PSLQ program is the best on this planet but with compound GAMMA values mixed with algebraic numbers, yes it can find some of the genuine Bill Gosper-like identities but for this you need more than brute force approach, you need a clue on the possible expressions because we are dealing here with non-linear expressions even if we take the log, unless you are mr Gosper itself, but mr Gosper is not a method it is a human being.!, ... There is so far no global answer to this problem, the best is a mix of all this I presume ? Best Regards to all, Simon Plouffe Le 29/12/2011 13:32, Robert Munafo a écrit :
On Wed, Dec 28, 2011 at 14:58, Warren Smith<warren.wds@gmail.com> wrote:
A certain proportion of Ries formulas seem to be "tasteless" and the sort of thing that are never going to happen, like using (e^pi) or 1/ln(pi) or e or sin(1) in an exponent...
Yes, definitely. As Bill Gosper just pointed out to me, sin(1) anywhere is practically useless. And e^pi is a curio at best ("Gelfond's Constant" [5]).
If you look at the webpage I wrote about ries [1], most of which is dedicated to "Stupid Math Tricks", perhaps the idea behind ries will be more clear.
I actually started by looking at all the stuff in Plouffe's Inverter, consisting mostly of long expressions using obscure and undefined functions, peppered with the occasional fraction like "2087/1457" which I could have figured out on my own. Then, by chance I got a couple emails in a row from members of the Cult of 137 [3], and I decided to prove to myself (and to them) that it is easy to make all sorts of meaningless formulas for any number you choose.
More to your point, the stuff on Plouffe's Inverter [4] is clearly all meaningful to someone, but most of it is of no meaning to any one person in particular, so we have a severe target marketing problem. Armed with these hundreds of higher-math functions, ries would be as bad, if not much worse. It will probably never have much serious application.
Nevertheless, it sure is a lot of fun, and fun is a large part of the meaning of life. So there, xyzzy!
- Robert.
[1] http://mrob.com/pub/ries/index.html
[3] http://mrob.com/pub/num/n-b137_035.html#cult_137
[4] http://bootes.math.uqam.ca/cgi-bin/ipcgi/lookup.pl?Submit=GO+&number=1.43239...