The problem-space seems much less interesting to me when you're allowed to have large coefficients on the powers of pi and/or e. The thing that makes pi^4+pi^5-exp(6) appealing is that the coefficients are all 1. We should at least place a bias against candidates with large coefficients by (say) multiplying the error term by all coefficients and using that as an "appeal factor" or "coolness" score. We would take the error term from this example: 44+1248*e^3 - 7993*pi = -5.016...x10^-9 and multiply it by 44*1248*7993 to get an "appeal factor" of 2.2019... Whereas the original pi^4 + pi^5 - exp(6) = -1.767...x10^-5 has all 1 coefficients and therefore an "appeal factor" of -1.767...x10^-5. Any automated search should be set up to focus its efforts on small coefficients (and small exponents too, if that turns out to matter) and sort its output by this adjusted "appeal factor". - Robert Munafo On 2012-04-29, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello,
actually, in these matters in fact, we can run a program to evaluate any expressions using 2 vectors of numbers like [powers of Pi] and [powers of e] and by tuning an integer relation algorithm to the proper Digits to find all of them in increasing size of approximation.
Here is the output after 10 minutes (and a little cleaning),
On each line the expression followed by the error in absolute value, we recognize the usual continued fraction approximations.
22 - 7 Pi, 0.008851424871447331
4 -5 -2143 + 22 Pi , 0.2748053619201687 10
3 -31 + Pi , 0.006276680299820175
4 -5 2143 - 22 Pi , 0.2748053619201687 10 [...] -8 44 + 1248 exp(3) - 7993 Pi, 0.5008014353103554 10
...
to discover some that are known too : like
4 5 -exp(6) + Pi + Pi , 0.00001767345123210921
The listing I have is 206000 lines, I stopped at 64 digits of precision. [...]
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com