A
search for a mathematical expression
for mass ratios using a large
database
Simon
Plouffe
preprint
Abstract
Using a
database of 610 millions mathematical constants and expressions a search was conducted
in order to find a reasonable expression based on simplicity and length for the
mass ratios of fundamental particles. The mass ratios are the dimensionless
values of the NISTCODATA 2002 values. The
search was restricted to 8 of the fundamental particles: the electron, proton,
neutron, helion, tau, deuteron, alpha and the muon since their mass is known to
enough precision.
The search
uncovered a weakness in a wellknown algorithm and produced a series of
candidate expressions that I propose as being the most appropriate mathematical
answer for that question. The paper explains the different models used.
Introduction
The mass ratio
of the proton and the electron is a dimensionless number, which is equal to
1836.15267261 with an uncertainty (85) on the last 2 digits. In other words, 9
digits are valid with a confidence of 99%.
In the 60’s,
Richard P. Feynman proposed the value of Mp/Me ≈ _{} = 1836.118107.
At the time it was considered like an inspired guess up to the known precision.
Compared to the known value today with the error it is not very good in fact
since the relative error 4.6 x 10^{6}
with (Mp/Me _{}) is 40663 times the error. A search using the author’s
database and programs with the other ratios failed to produce similar formulas in
simple terms of π^{k}, exp(πk) with k being an
integer or even rational.
That idea of
finding the best mathematical expression for dimensionless numbers of physics
goes back to Sir Arthur Eddington in the 30’s with Einstein at the beginning, later Feynman, Gellman and Dirac addressed or
thought of that problem. There is an extensive literature on the subject.
The project was
to use a large database of mathematical constants and specialized programs to
find an expression for many if not all
mass ratios. The previous results like the one of Feynman were more or less
guessed and I believe that this method was inspired but not appropriate.
The Inverter and methodology
The Inverter is
the private version of Plouffe’s Inverter found on the internet with over 610
million of mathematical constants including the 99000 sequences of the Online Encyclopaedia of Integer sequences
of [Sloane] and a subset of 797 mathematical constants from Steven Finch’s
Mathematical Constants [Finch]. The constants are values of known functions or
series at rational points, known constants like gamma, π, √2 or the
golden ratio. The construction of the tables were inspired from known
references like [AS][Boll][Le
Lionnais][PotterRobinson], [Sloane, Plouffe] and [Finch].
Each sequence
of the OEIS is transformed into a real number entry using various power series
representations, continued fractions and concatenations.
The existence
of the database is based on the assumption that a number like 0.42278433 can
only be recognized or known if it is known by advance that it is 1gamma. Gamma
(0.5772156649…) is a known constant to mathematics but not that well known
otherwise and there is no simple way to go from 0.422… to 1gamma. As of today there are algorithms that can find
that value but they are not simple. Such algorithms like the Integer Relations
algorithm, the LLL algorithm (Lenstra, Lenstra, Lovasz)
or the PSLQ algorithm of Bailey and
Once the
database is set with all known sources of mathematical constants a global
lookup was conducted with the CODATA table at precisions varying from 5 to 11
digits. A simple criteria based on the length in characters of the expression
associated with each number is used to classify candidates.
To catch more
possible candidates the Farey fractions (Farey(n) is a
set of rationals in ]0,1] with denominators ≤ n), of order 12, 24, 60,
120, 240, 256 and elementary transforms and functions on each entry is used for
lookups as well. For each of the CODATA entry a table was produced and sorted
in increasing length of expressions, a tree shaped table having on top the
simplest expression associated with a value. These calculations produced a set
of 23 million candidates that were compared and analyzed to find similarities
among tables. No definite pattern emerged from that analysis and on top of all
trees the simplest expression found was Mp/Mn ≈ cos(p/60), valid to
5 digits. That value is elegant and simple but unique among candidates and not
precise enough, the difference with the real value is more than 10,000 times
the standard error.
Integer Relations algorithm
Many
names are known for the next algorithms. Usually they refer to Integer Relation
algorithms and can be stated in the following way. Let x =
(x_{1}, x_{2}, x_{3}, …, x_{n}) be a vector of
real numbers. x is said to possess an integer relation if there exist integers
not all zero such that a_{1}x_{1} + a_{2}x_{2} + … + a_{n}x_{n }=
0. By an integer relation algorithm, I mean an algorithm that is guaranteed
(provided the computer implementation has sufficient numerical precision) to
recover the vector of integers a_{i}, if it exists, or to produce
bounds within which no integer relation can exist.
The
2 dimension version of the algorithm is the continued fraction algorithm and is
equivalent to the
These programs
can easily attack problems dealing with hundreds of decimal digits but are
almost useless for small problems dealing with only 5 because of the error
control. Usually the safe bounds for the error control is
handled with the double of the precision but with 5 digits the answer provided is
hardly valid. In the best cases, 11 digits are just enough but with the known
error bound for the constants the results are poor. Nevertheless an extensive
search was conducted using models like a combination of π, exp(π), exp(2 π) and such and some individual
results were found. But when an individual result is found that cannot be
reproduced for the other ratios then it must be rejected because it has no
value apart from being a numerical curiosity. For powers of π alone, 1 million
expressions were found similar to Feynman expression but no pattern emerged as
well.
electron 
9.1093826e31 

muon 
1.8835314e28 

proton 
1.67262171E27 

neutron 
1.67492728E27 

tau 
3.16777e27 

deuteron 
3.34358335e27 

helion 
5.00641214e27 

alpha 
6.6446565e27 

neutronproton 
1.00137841870 

tauneutron 
1.89129000000 

tauproton 
1.89390000000 

deuteronproton 
1.99900750082 

helionproton 
2.99315266710 

alpha
particleproton 
3.97259968907 

protonmuon 
8.88024333000 

neutronmuon 
8.89248402000 

taumuon 
16.81830000000 

muonelectron 
206.76828380000 

protonelectron 
1836.15267261000 

neutronelectron 
1838.68365980000 

tauelectron 
3477.48000000000 

deuteronelectron 
3670.48296520000 

helionelectron 
5495.88526900000 

alpha
particleelectron 
7294.29953630000 

Masses in KG and ratios from the NISTCODATA 2002
table
Notes
: Only the values > 1 are taken into account and
only if they appear separately as a ratio in the CODATA 2002 table. The ratios
are the results of many experiments and averages and not the result of the
arithmetical operation of taking ratios of entries in the first table
Model
#1, spheres or archimedean solid of n dimensions
Ndimensional
spheres of uniform matter is the first model considered. The
volume of a ndimensional sphere V(n) is _{}, consequently the mass ratio should be rational in 3
dimensions and with powers of π if the dimension is higher. I could not
find any evidence of such hypothesis. The next step is to consider semiregular
polytopes such as the archimedean solids. The volume of such polytopes is expressed
in radicals so I expected the mass ratios to be as well.
(Courtesy of Eric Weisstein from Math
World at Wolfram Research:
http://mathworld.wolfram.com/ArchimedeanSolid.html).
If I go back to
the number cos(π/60), that number is algebraic
and the value is
cos(π/60)
= _{}
This is a fairly simple expression and
could represent a ratio of such polytopes but a problem arises when I consider
the polynomial which has this number as a root. The degree is 24, cos(π/60) is one of the roots of
_{}
I can’t use the known 11 digits of
Mp/Mn to find such a polynomial using PSLQ or LLL algorithms. The only way to
detect simple expressions with radicals with as little as 11 digits is to
construct tables of values like cos(π/60),
values of algebraic numbers with embedded radicals, roots of simple polynomials
and combination of roots of unity. In all, there are 245 million algebraic
entries of various degrees in the main table. I applied the same method and
found nothing simpler (but more precise) than the cos(π/60)
expression.
Model
#2 expressions with p,
exp(p) and various bases.
I go back to Feynman and also that z(n) and/or powers of p, consider these products.
_{}
n is integer, p is a prime ≥
2 and c is a composite number.
The product wth the inverses of all
primes is expressed with p^{4} and the product of inverses of all
integers is related to p/exp(p), therefore the product of
inverses of all composite numbers is simply the ratio. In other words when I
have an expression with p^{2 }and 1/p^{2 }then it means something if I
have Mp/Me ≈ 6p^{5}+ 328/p^{8} then it hardly can be
explained in terms of primes and composites, the exponents have to be related
in some way. Other bases like Fibonacci and φ were tried, p bases and exp(p) bases as well. In all
cases it had to correspond to a pattern, a similarity in either the exponent or
the coefficient. Unfortunately no patterns were found despite the numerous
candidates.
Here is a
summary of the models with the number of entries and method used.
Type of model 
Expected ratio type 
Method of detection 
Number of entries in main table 
Best possible match with any entry 
Any from shortest to longest expression 
Lookups + variations 
610 million entries 
Sphere in n dimension 
Powers of p with simple algebraic numbers. 
Integer Relations or LLL 
Dynamic lookup 
Archimedean solid of n dimension 
Algebraic numbers with embedded radicals. 
Lookups with variations 
145 million entries 
Additive with single base like exp(Pi*k), X^{k} or roots of unity. 
Linear combination of base. 
Integer Relations or LLL and GFUN. 
Dynamic lookup 
Related to a specific integer sequence eveluated at ±1,
exp(p) or exp(2p). 
Any from shortest to longest expression * 
GFUN, LLL, lookups + variations. 
13 million entries 
Single constant generation X 
Multiplicative with powers of X : A/B * X^{k} 
Construction of specialized table + lookups with
variations. 
145 million entries 
Families of
expressions found for many ratios
These are the near identities involved with F(n),
L(n) and φ. F_{n} are the Fibonacci numbers : 1,
2, 3, 5, 8, 13, 21, 34, 55, … = the
nearest integer to φ^{n}/√5. L_{n }are the Lucas numbers : 2, 3, 4,
7, 11, 18, 29, 47, 76, 123, … = the
nearest integer to φ^{n} .
By using the definition of those numbers, I can deduce 3 basic transformations
that will lead to a nearby value.
_{}
And
from there another series of transforms when r divides p, a
and b being integers.
_{}
This
last set of identities that are almost equal to 1 forced me to reconsider many
expressions encountered especially one found about the alpha particle and
electron ratio, that is
_{} valid
up to 7 digits
Since 11 ≈ φ^{5}
, 11 is the 5’th Lucas number and 29 is ≈ φ^{7} this
identity is in fact a power of φ in disguise. This is a very simple
expression. But since I have 3 different expressions near 1 it means that in
fact there is a set of values near that point. This is no surprising that I
could not find it with those Integer Relations algorithms since it is linear
with [log(φ), log(Malpha/Melectron), log(11),
log(29)] but since the numeric precision is only 11 digits at the most then the
algorithm fails to find it. Ihad to construct a specialized table of 145
million entries to be sure to detect any of these relations and this is what I have
found.
But since each expression can be either
improved in precision or simplified then it means that for each approximation
there is a family of expressions near the value, an infinite family of
expressions all similar. This is by far the simplest model I have found. Each
expression is generated by 1 single number at a given power,
that is the golden ratio.
Those ripples appear in group near the
ratio values of the CODATA 2002, in some cases I could find families of
families of values like
_{}.
These patterns
can be explained easily since for each n the basic transformation will lead to
a series of near values once transformed. That phenomena
explains why I have those series.
_{}, _{},_{},_{} ≈ 1.00137841870
Summary
of results for φ = ½+√5/2 = 1.6180339887…
Particle Ratio 
Value 
Simplest Expression 
Other expressions 
Other expressions 
NeutronProton 
1.00137841870 
_{} 
_{} 
_{} 
TauProton 
1.8939 
_{} 


TauNeutron 
1.89129 
_{} 
_{} 
_{} 
HelionProton 
2.9931526671 
_{} 
_{} 

AlphaparticleProton 
3.97259968907 
_{} 
_{} 
_{} 
NeutronMuon 
8.89248402 
_{} 


ProtonMuon 
8.88024333 
_{} 


TauMuon 
16.8183 
_{} 
_{} 

MuonElectron 
206.7682838 
_{} 
_{} 

ProtonElectron 
1836.15267261 
_{} 
_{} 

NeutronElectron 
1838.6836598 
_{} 
_{} 
_{} 
DeuteronElectron 
3670.4829652 
_{} 


HelionElectron 
5495.885269 
_{} 


Alpha
particleElectron 
7294.2995363 
_{} 


Error analysis
Particle Ratio 
Value 
Best Expression 
Error 
Relative
error 
NeutronProton 
1.00137841870 
_{} 
0.1375 x 10^{8} 
2.374·e 
TauNeutron 
1.89129 
_{} 
0.48467 x10^{4} 
1.573·e 
TauProton 
1.8939 
_{} 
0.34040 x 10^{5} 
0.01098·e 
HelionProton 
2.9931526671 
_{} 
0.36516x10^{7} 
6.2959·e 
AlphaparticleProton 
3.97259968907 
_{} 
0.9035 x 10^{10} 
0.17376·e 
ProtonMuon 
8.88024333 
_{} 
0.2782 x 10^{7} 
0.12098·e 
NeutronMuon 
8.89248402 
_{} 
0.15001 x 10^{6} 
0.6522·e 
TauMuon 
16.8183 
_{} 
0.3754 x 10^{7} 
0.0000139·e 
MuonElectron 
206.7682838 
_{} 
0.149 x 10^{4} 
2.764·e 
ProtonElectron 
1836.15267261 
_{} 
0.22071 x 10^{5} 
2.596·e 
NeutronElectron 
1838.6836598 
_{} 
0.17397 x 10^{6} 
0.1338·e 
DeuteronElectron 
3670.4829652 
_{} 
0.1192 x 10^{5} 
0.6625·e 
HelionElectron 
5495.885269 
_{} 
0.2844 x 10^{2} 
2585·e 
Alpha
particleElectron 
7294.2995363 
_{} 
0.38535 x 10^{3} 
120·e 
3 expressions are out of the range of the normal error
(in gray) and
have to be rejected. All the others are within the normal error,
that is +/ 3e.
Conclusion
Now the
questions is are these patterns occuring with any real number or is it occuring
especially for those ratios? By using an algorithm to systematically replace an
expression by a nearby expression with the basic transforms then I get roughly 1
digit of precision by iteration (or term). In other words a 10 digits
approximation of an arbitrary real number will lead to an expression with 10
terms and by looking at the size of the expressions obtained then I conclude
that they are remarkable and that these
ripples of values appear near the
values of the CODATA 2002 and they fit within the error bounds. The expressions
are in my opinion the simplest mathematical expression that can exist for those
numbers.
As I mentioned the problem is not to
find an answer but to find an answer for the 16 ratios that makes sense and
above all a comprehensive or simple answer if there is any. After all, those
ratios could vary with time and be not constant at all as suggested by recent
findings. Even Paul A.M. Dirac doubted that any mathematical expression could
even exist.
A
weakness discovered in Integer Relations algorithms when using a small
precision.
In the course of experiments I dealt with one simple
case that is an integer relation with 1,
√5 and φ^{48} . As
you may know the golden ratio has many facets and one of them is the relation
φ^{n} =F(n)φ+F(n1), with F(n) being
the n’th Fibonacci number. But also that φ^{n}
is very near integers when n >> 1. We expect the program to at least
detect that but it is not the case since when asked to solve in integers [1,
√5, φ^{48}] it answers [1791659574,
1, 4006272456] when the actual answer is [5374978561, 2403763488, 1] that is,
φ^{48 }is a linear combination of F(48)√5
and F(47). It does it well if Iincrease the number of working digits to 100 but
that mises the point when the digits are set to 24. This is exactly why Icould
not rely on that algorithm to find valid integer relations with a working
precision that goes from 5 to 11 digits.
References
[Aspden, Eagles] H. Aspden and D. M. Eagles, Physics Letters A, v. 41, p.423
(1972).
[AS] Abramowitz, M. and Stegun,
[Bailey,
[Bergeron, Plouffe] Computing the
generating function of a series given its first few terms, Experimental Mathematics,
Vol. 1 #4, 1992.
[Boll] Marcel Boll, Tables Numériques Universelles des laboratoires et bureaux d'étude, 881 pages. Dunod
1947
[CSD] JeanPaul
Delahaye, Certitudes sans démonstrations, in Pour La Science, #249
, juillet 1998. See article :
http://www.lacim.uqam.ca/~plouffe/articles/Certitude_sans_demonstration.pdf
[Finch] S.
Finch, Mathematical Constants,
Encyclopedia of Mathematics and its Applications,
[GFUN] GFUN a maple package for
manipulating power series, Bruno Salvy and Paul Zimmermann, Paris 1992, INRIA
internal report.
[Hardy, Wright] Hardy, G. H. and Wright,
W. M.
An Introduction to the Theory of Numbers, 5th ed.
[Hastad et al] J. Hastad, B. Just, J. C. Lagarias and C. P.
Schnorr,Polynomial Time Algorithms for Finding Integer
Relations Among Real Numbers,"
[Le Lionnais] Les nombres
remarquables, Paris, Hermann, 1983.
[LLL] A. K. Lenstra, H. W. Lenstra
and L. Lovasz, Factoring Polynomials with
Rational Coeffcients, Math. Annalen, vol. 261 (1982), p. 515  534.
[NIST CODATA 2002] http://physics.nist.gov/cuu/Constants/Table/allascii.txt
[Plouffe’s Inverter] http://pi.lacim.uqam.ca/eng/.
[Inverseur de Plouffe] http://pi.lacim.uqam.ca/fra/.
{FergusonForcade], Generalization
of the Euclidean Algorithm for Real Numbers to All Dimensions Higher Than Two,"
Bulletin of the
American Mathematical Society, 1 (1979), p. 912  914.
[PotterRobinson] copy of a
manuscript document from Jeff O. Shallit 1995.
[PSLQ]
[Sloane, Plouffe] The encyclopedia of Integer
Sequences, Academic Press,
[Sloane N.J.A.] The
OnLine Encyclopedia of Integer Sequences.
http://www.research.att.com/~njas/sequences/.
[MohrTaylor] Peter J. Mohr and Barry N. Taylor, CODATA Recommended Values of the Fundamental Physical Constants: 2002 (to appear). Latest version is CODATA of Dec. 2003
See also : http://physics.nist.gov/constants