[math-fun] pi^4 + pi^5 ~ e^6
On 2012-04-27, David Wilson <davidwwilson@comcast.net> wrote:
Yes, and then there was
pi^4 + pi^5 = e^6
which was found by me, however, I found it long ago in an email signature on a USENET group.
I was impressed, and later mentioned it to Eric Weisstein, and it ended up in MathWorld (Almost Integer) crediting me.
From there it went to Wikipedia (Mathematical Coincidence).
Does it have a prior history in the literature?
My entry on the number has a footnote giving this page: http://zhurnaly.com/cgi-bin/wiki/CoincidentalTaxonomy which (after the article but before the comment section) says "TopicScience - Datetag20011019", suggesting a date of Oct 19, 2001. -- 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
This reference probably predates my discovery of the near-identity, however, I seem to think I knew about it in early 2002 when I lost my job at Cabletron. Whereas I know I didn't author it myself, so I'm slightly uncomfortable with the MathWorld credit if an earlier source can be uncovered. Back when Eric was highly involved in MathWorld, it was easy for me to make changes to it, now the process seems rather daunting. Sort of like Wikipedia, the last three significant updates I made to Wikipedia were eventually reverted by self-proclaimed keepers of the articles at hand, so I gave up on contributing to Wikipedia. "The encyclopedia that anyone can edit" is technically true, with the understanding that anything you contribute will ultimately be removed. On 4/27/2012 8:21 PM, Robert Munafo wrote:
On 2012-04-27, David Wilson<davidwwilson@comcast.net> wrote:
Yes, and then there was
pi^4 + pi^5 = e^6
which was found by me, however, I found it long ago in an email signature on a USENET group.
I was impressed, and later mentioned it to Eric Weisstein, and it ended up in MathWorld (Almost Integer) crediting me.
From there it went to Wikipedia (Mathematical Coincidence).
Does it have a prior history in the literature? My entry on the number has a footnote giving this page:
http://zhurnaly.com/cgi-bin/wiki/CoincidentalTaxonomy
which (after the article but before the comment section) says "TopicScience - Datetag20011019", suggesting a date of Oct 19, 2001.
David, you must have told Eric Weisstein about it at least as early as 1998. I just checked my copy of "The CRC concise encyclopedia of mathematics", dated 1998 (CRC Press, ix + 1969 pages, LC number 98-22385). It includes the near-identity and credits "D. Wilson". It is on page 33, right column, equation (7) near the bottom. On 2012-04-28, David Wilson <davidwwilson@comcast.net> wrote:
This reference probably predates my discovery of the near-identity, however, I seem to think I knew about it in early 2002 when I lost my job at Cabletron. Whereas I know I didn't author it myself, so I'm slightly uncomfortable with the MathWorld credit if an earlier source can be uncovered.
Back when Eric was highly involved in MathWorld, it was easy for me to make changes to it, now the process seems rather daunting.
Sort of like Wikipedia, the last three significant updates I made to Wikipedia were eventually reverted by self-proclaimed keepers of the articles at hand, so I gave up on contributing to Wikipedia. "The encyclopedia that anyone can edit" is technically true, with the understanding that anything you contribute will ultimately be removed.
On 4/27/2012 8:21 PM, Robert Munafo wrote:
On 2012-04-27, David Wilson<davidwwilson@comcast.net> wrote:
Yes, and then there was
pi^4 + pi^5 = e^6
which was found by me, however, I found it long ago in an email signature on a USENET group.
I was impressed, and later mentioned it to Eric Weisstein, and it ended up in MathWorld (Almost Integer) crediting me.
From there it went to Wikipedia (Mathematical Coincidence).
Does it have a prior history in the literature? My entry on the number has a footnote giving this page:
http://zhurnaly.com/cgi-bin/wiki/CoincidentalTaxonomy
which (after the article but before the comment section) says "TopicScience - Datetag20011019", suggesting a date of Oct 19, 2001.
-- 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
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. On 4/28/2012 11:24 AM, Robert Munafo wrote:
David, you must have told Eric Weisstein about it at least as early as 1998.
I just checked my copy of "The CRC concise encyclopedia of mathematics", dated 1998 (CRC Press, ix + 1969 pages, LC number 98-22385). It includes the near-identity and credits "D. Wilson". It is on page 33, right column, equation (7) near the bottom.
On 2012-04-28, David Wilson<davidwwilson@comcast.net> wrote:
This reference probably predates my discovery of the near-identity, however, I seem to think I knew about it in early 2002 when I lost my job at Cabletron. Whereas I know I didn't author it myself, so I'm slightly uncomfortable with the MathWorld credit if an earlier source can be uncovered.
Back when Eric was highly involved in MathWorld, it was easy for me to make changes to it, now the process seems rather daunting.
Sort of like Wikipedia, the last three significant updates I made to Wikipedia were eventually reverted by self-proclaimed keepers of the articles at hand, so I gave up on contributing to Wikipedia. "The encyclopedia that anyone can edit" is technically true, with the understanding that anything you contribute will ultimately be removed.
On 4/27/2012 8:21 PM, Robert Munafo wrote:
On 2012-04-27, David Wilson<davidwwilson@comcast.net> wrote:
Yes, and then there was
pi^4 + pi^5 = e^6
which was found by me, however, I found it long ago in an email signature on a USENET group.
I was impressed, and later mentioned it to Eric Weisstein, and it ended up in MathWorld (Almost Integer) crediting me.
From there it went to Wikipedia (Mathematical Coincidence).
Does it have a prior history in the literature? My entry on the number has a footnote giving this page:
http://zhurnaly.com/cgi-bin/wiki/CoincidentalTaxonomy
which (after the article but before the comment section) says "TopicScience - Datetag20011019", suggesting a date of Oct 19, 2001.
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.
-- 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
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.
Hi, thanks for the idea.... I decided to try: z^n + z^(n-1) - pixel^(n+1) As a standard escape-time formula - very nice results :) e.g. try plain z^2+z-pixel^3 with a start of (-0.5,0). Also the reverse works nicely to, either as: z^n - sigma(pixel^p) where p is either from 0 to n-1 or from 1 to n-1 The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality.
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
I did not peruse rec.arts.sf-lovers, so I would likely have seen Doug Ingram's followup message. I'm sure I told Eric Weisstein back when he was working on the Treasure Troves of Science, which is how it would have found its way into the CRC Encyclopedia of Mathematics, and Wolfram MathWorld. I have never heard anything about it in the literature prior to 1989, nor anything more about it from the presumed discoverer. All subsequent references could reasonably be traced back to me. So I guess that if I had not told Eric about it, this little nugget might have been lost. It would be nice to see the MathWorld article updated to acknowledge the likely source. Nice piece of detective work, btw. On 4/29/2012 8:10 AM, Robert Munafo wrote:
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.
participants (4)
-
Dave Makin -
David Wilson -
Robert Munafo -
Simon Plouffe