mass ratio of neutron to proton
plouffe
I have been working on a suitable math expression for the
constant : neutron to proton mass ratio. According to the latest
results in http://physics.nist.gov/cuu/Constants/Table/allascii.txt
FIrst, there is a inherent limitation to these constants since the
new definition of the speed of light and the second, in other
words we are limited to a precision of 10-11 digits in all best
cases.
Nevertheless, I took the challenge and liberty of trying to find
a suitable math expression over the past years using the tables
of the inverter and a couple of techniques with pari-Gp, the
LLL algorithm or the PSLQ algorithm still with only 11 digits.
The best and simplest expression I could find is the following
1/2
40 1 8 3
-- --------- - ------
27 Pi 27
cos(----)
15
or when line printed : 40/27/cos(1/15*Pi)-8/27*3^(1/2)
or 8/27 * (5/cos(Pi/15) - sqrt(3))
This is an interesting one for more than 1 reason,
first the cos(Pi/15) can be simplified to
1/2 1/2 1/2 1/2 1/2
2 3 (5 + 5 ) 5
----------------------- + ---- - 1/8
8 8
perhaps this can be simplified further.
Now the decimal expansion of that number is
1.001378419779635 when the REAL value is
1.00137841918...
this means that the error is 1.28e where e
is 0.000 000 000 46 I think that this is an acceptable error.
This is only a guess based on a large database (2.459 billion ) of
mathematical constants and program I developed
, I do not pretend it could
be the real value. Perhaps it could be interesting to
see IF that value can be obtained VIA Ramanujan Continued fractions
like : http://en.wikipedia.org/wiki/Ramanujan_continued_fractions
or see http://citeseer.ist.psu.edu/249592.html and click on the PDF
version or
the Ramanujan Notebooks 5 page 9 and following pages...
actually the whole book is quite interesting too!
Can someone find a better expression ?
short, elegant and simple that FITS the error
within an acceptable limit ?
Simon Plouffe
Bluuuuue Rajah
4e4528...@e39g2000hsf.googlegroups.com:
> Hello,
>
> I have been working on a suitable math expression for the
> constant: neutron to proton mass ratio. According to the latest
> results in http://physics.nist.gov/cuu/Constants/Table/allascii.txt
>
> First, there is a inherent limitation to these constants since the
> new definition of the speed of light and the second, in other
> words we are limited to a precision of 10-11 digits in all best
> cases.
Since the proton and neutron are of very similar mass, you can save
yourself some accuracy by looking for the expression 1-mp/mn or mp/mn-1.
Then you'll probably only need the traditional 3 decimal places,
although you can add as much as you want to that, at your whim.
Of course, protons and neutrons are also known to consist of up and down
quarks, so you may want to express your answer as the ratio of quark
masses, rather than protons and neutrons, the latter being so very
1960's. ;)
> Nevertheless, I took the challenge and liberty of trying to find
> a suitable math expression over the past years using the tables
> of the inverter and a couple of techniques with pari-Gp, the
> LLL algorithm or the PSLQ algorithm still with only 11 digits.
>
> The best and simplest expression I could find is the following
>
> 1/2
> 40 1 8 3
> -- --------- - ------
> 27 Pi 27
> cos(----)
> 15
> or when line printed : 40/27/cos(Pi/15)-(8/27)*3^(1/2)
>
> or (8/27)^3 * (5/cos(Pi/15) - sqrt(3))
Notice that 8/27 is (2/3)^3.
> This is an interesting one for more than 1 reason,
> first the cos(Pi/15) can be simplified to
>
> 1/2 1/2 1/2 1/2 1/2
> 2 3 (5 + 5 ) 5
> ----------------------- + ---- - 1/8
> 8 8
>
> perhaps this can be simplified further.
You can pull out a factor of 1/8, and get
1/2 1/2 1/2 1/2 1/2
( 2 3 (5 + 5 ) + 5 - 1 ) / 8.
> Now the decimal expansion of that number is
> 1.001378419779635 when the REAL value is
> 1.00137841918...
> this means that the error is 1.28e where e
> is 0.000 000 000 46 I think that this is an acceptable error.
Count zeroes to get 0.46 x 10^-9, which is about 1/2 part per billion.
> This is only a guess based on a large database (2.459 billion ) of
> mathematical constants and program I developed, I do not pretend it
> could be the real value.
You have tested your theory against experiment, which I think is all
that anyone can reasonably ask. You may have a good publication here,
or at least a good conference presentation. You should also give a
colloquium at your local university.
You should track down some other people who understand those three
algorithms you mentioned, and get them to be your referees. An error of
0.46 in a billion would seem to be quite reliable, but of course, this
sort of thing always has to be vetted by another expert in the field, to
make sure you haven't missed something simple.
Do you have a detailed writeup of all the important steps you used in
your derivation?
> Perhaps it could be interesting to see IF that value can be obtained
> VIA Ramanujan Continued fractions like:
> http://en.wikipedia.org/wiki/Ramanujan_continued_fractions
> or see http://citeseer.ist.psu.edu/249592.html and click on the PDF
> version or the Ramanujan Notebooks 5 page 9 and following pages.
> Actually the whole book is quite interesting too!
>
> Can someone find a better expression - a short, elegant and simple
> that FITS the error within an acceptable limit?
This is not my field of expertise, but if you haven't made any algebra
mistakes, you would seem to be at the state of the art. It's time to
take this to the next level, and publish or perish. There's a rumor
going around that amateur science is a movement that's taking on
momentum. Your work may be a great example of this. :)
Peter Webb
"plouffe" <simon....@gmail.com> wrote in message
news:8f46937b-69a9-40aa...@e39g2000hsf.googlegroups.com...
> Hello,
>
> I have been working on a suitable math expression for the
> constant : neutron to proton mass ratio. According to the latest
> results in http://physics.nist.gov/cuu/Constants/Table/allascii.txt
>
> FIrst, there is a inherent limitation to these constants since the
> new definition of the speed of light and the second, in other
> words we are limited to a precision of 10-11 digits in all best
> cases.
> Nevertheless, I took the challenge and liberty of trying to find
> a suitable math expression over the past years using the tables
> of the inverter and a couple of techniques with pari-Gp, the
> LLL algorithm or the PSLQ algorithm still with only 11 digits.
>
> The best and simplest expression I could find is the following
>
>
> 1/2
> 40 1 8 3
> -- --------- - ------
> 27 Pi 27
> cos(----)
> 15
> or when line printed : 40/27/cos(1/15*Pi)-8/27*3^(1/2)
>
> or 8/27 * (5/cos(Pi/15) - sqrt(3))
>
Counting "cos", "sqrt", PI and each operator(+-/*) as a single byte each,
and also counting each digit in each number as a byte, I count this as 15
bytes long.
> This is an interesting one for more than 1 reason,
> first the cos(Pi/15) can be simplified to
>
> 1/2 1/2 1/2 1/2 1/2
> 2 3 (5 + 5 ) 5
> ----------------------- + ---- - 1/8
> 8 8
>
> perhaps this can be simplified further.
>
> Now the decimal expansion of that number is
> 1.001378419779635 when the REAL value is
> 1.00137841918...
These agree to 11 bytes.
So the formula giving the number is longer than the number itself.
> this means that the error is 1.28e where e
> is 0.000 000 000 46 I think that this is an acceptable error.
>
Is it? How well known is the exact value?