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 9 -29809 + Pi , 0.09933344621166651 5 -306 + Pi , 0.01968478528145326 -355 + 113 Pi, 0.00003014435336405372 10 -93648 + Pi , 0.04747608302097372 2 227 - 23 Pi , 0.0009012250552482332 2 -10975 + 1112 Pi , 0.00009401136678414395 -20 + exp(3), 0.08553692318766774 -8103 + exp(9), 0.08392757538400771 -2981 + exp(8), 0.04201295827172526 6 -306 Pi + Pi , 0.06184157682770606 4 -31 Pi + Pi , 0.01971877271884684 2 -79 + 8 Pi , 0.04316479128513105 6 -5 1266 - 709 Pi + Pi , 0.2180141030960205 10 5 17 + 92 Pi - Pi , 0.006839344979524676 2431 - 329 exp(2), 0.0005434518160752412 12 -294204 Pi + Pi , 0.05646664265629392 2 5 -31 Pi + Pi , 0.06194835151133608 2 -355 Pi + 113 Pi , 0.00009470127907572594 133 - 18 exp(2), 0.003009780751704090 19 - 7 exp(1), 0.02797279921331665 4 9 2 + 306 Pi - Pi , 0.08252295853412784 5 8 2 + 31 Pi - Pi , 0.07921227315104402 2 -6 2195 + 71 Pi - 245 Pi , 0.1379824583163974 10 2 22 Pi - 7 Pi , 0.02780757134994091 5 8 -5 258 - 776 Pi + 25 Pi , 0.8385942446327131 10 3 4 355 Pi - 113 Pi , 0.0009346641607545763 6 9 6 + 31 Pi - Pi , 0.03433261177411857 3 6 961 - 62 Pi + Pi , 0.00003939671558615069 -1457 + 536 exp(1), 0.0009399459517538469 -193 + 71 exp(1), 0.001990179407788289 2 -35 + 8 Pi + Pi , 0.002345629807704527 6 11 19 + 306 Pi - Pi , 0.07526015256062555 7 10 19 + 31 Pi - Pi , 0.04258499753311922 3 5 213 + 3 Pi - Pi , 0.0008547443819927363 2 3 22 Pi - 7 Pi , 0.08736006186714839 3 227 Pi - 23 Pi , 0.002831282012798905 241 - 12 exp(3), 0.02644307825201289 3 25 + 298 Pi - 31 Pi , 0.00003368046395962210 4 -6 2702 + 4442 Pi - 171 Pi , 0.4314447978197650 10 9 2738 + 8617 Pi - Pi , 0.004562537036669323 3 28 - 78 Pi + 7 Pi , 0.0002902179051313718 2 4 3 - 99 Pi + 10 Pi , 0.00007463217786909979 4 5 355 Pi - 113 Pi , 0.002936334061000247 -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. In more general terms, one can do that with any constants and see if for example the values of Zeta(n), n>1 are well approximated by exponentials, or any other function. Best regards, Simon PLOUFFE Le 29/04/2012 14:10, Robert Munafo a écrit :
I traced it back to apparently the originator of the .sig file:
http://groups.google.com/group/rec.arts.sf-lovers/browse_thread/thread/639d7...
I've added this to my entry on pi^4+pi^5:
mrob.com/pub/math/numbers-12.html#lb403_428
Original USENET messages follow:
----8<--snip-here---- Newsgroups: rec.arts.sf-lovers From: so...@ohstpy.mps.ohio-state.edu (Soren G. Frederiksen -- Ohio State University) Date: 3 Jul 89 12:22:36 GMT Local: Mon, Jul 3 1989 8:22 am Subject: Piers Anthony
In article<2...@maytag.waterloo.edu>, gigu...@aries5.uucp (Eric Giguere) writes:
One thing I find very interesting about Piers Anthony is his penchant for multi-volume series. Now I don't really mind this, but there's a problem: the first two or three books in the series (or maybe just the first) are very enjoyable, but things taper off from there. As examples consider "On a Pale Horse" and "A Spell for Chameleon". But the latest Xanth novels (what's he up to now? 10? 11?) and the last three in the Incarnations series haven't been as good as the series openers, at least in my opinion. I agree with you to a certain extent, I prefer the first few books of the Xanth series much more than the later books. However, personally I think that the first and the last books in the Incarnations series are the best, which is interesting considering that the sixth book seems to be more of an after thought. ---- Hmmmm, I wonder if that says something about my or Piers Anthony's personality in that the best books are about death and evil??? Soren Frederiksen
4 5 6 PI + PI = e ????? Strange enough to be true.
Newsgroups: rec.arts.sf-lovers From: v...@unix.cie.rpi.edu (VICC Project (Rose)) Date: 3 Jul 89 17:57:22 GMT Local: Mon, Jul 3 1989 1:57 pm Subject: Re: Piers Anthony
In article<2...@ohstpy.mps.ohio-state.edu>, so...@ohstpy.mps.ohio-state.edu (Soren G. Frederiksen -- Ohio State University) writes:
4 5 6 PI + PI = e ????? Strange enough to be true.
Not quite according to my HP41 but close enough to be interesting. I get 6 403.4287935 for e and 4 5 403.4287761 for PI + PI Frank Filz ----8<--snip-ends----
On 2012-04-28, David Wilson<davidwwilson@comcast.net> wrote:
I first started working on the OEIS back in 1997 (My earliest contributions were in the A02xxxx range). I'm pretty sure I knew about the identity at that time, and I guess this clinches it.
But still, it came from an email signature in a Usenet group, so perhaps we could find the originator in the Usenet archives, if there is such a thing.