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Exact mathematical expressions of the Proton to Electron Mass Ratio with symmetrical shape and physical meaning
Stergios Pellis
15/10/2021
Greece
Abstract
We present 26 exact mathematical expressions of the Proton to Electron Mass Ratio with exact value μ=1.836,15267343. We propose one exact mathematical expression using Fibonacci and Lucas numbers:
µ32=φ-42·F5160·L547·L1940/19
Also we present exact mathematical expressions between the Proton to Electron Mass Ratio and mathematical constants and also exact mathematical expressions between the Proton to Electron Mass Ratio and the Fine Structure constant α.
1. Introduction
The proton-to-electron mass ratio μ is a ratio of like-dimensioned physical quantities,it is a dimensionless quantity,a function of the dimensionless physical constants,and has numerical value independent of the system of units. Two of the great mysteries of physics are the origin of mass and the mysterious mass ratio between the proton and electron.
The numerical challenge of the mass ratio of proton to electron in the field of elementary particle physics began with the discovery of the electron by JJ Thomson in 1.897,and with the identification of the point nature of the proton by E. Rutherford in 1.911. These two particles have electric charges that are identical in size but opposite charges.
2. Measurement of the Proton to Electron Mass Ratio
The 2.018 CODATA recommended value of μ is:
μ=1.836,15267343
With standard Uncertainty 0,00000011 and Relative Standard Uncertainty 6,0×10-11. The value of μ is known at about 0,1 parts per billion. The value of μ is a solution of the equation:
3·μ4-5.508·μ3-841·μ2+10·μ-2.111=0
The 2.018 CODATA recommended value of μ-1 is:
μ-1=me/mp=0,000544617021487
With standard Uncertainty 0,000000000000033 and Relative Standard Uncertainty 6,0×10-11.
3. Background of the search for mathematical expression
The search for mathematical expression for this dimensionless number motivated many serious scientists. First Peirles in 1.928 proposed the mathematical expression:
μ=2·(π-1)·π/α
A year later Reinhold Furth in 1.929 assumed that μ could be derived from the quadratic equation containing the Fine Structure constant α:
μ=64·π/15·α
Later in 1.935,A. Eddington,who accepted some of Furth's ideas,presented in his book «New Pathways in Science» the equation for the the Proton to Electron Mass Ratio μ:
10·μ2-136·µ+1=0
However both approaches can not be used nowadays as they give very high deviation from the currently known experimental value of µ,so they are not reviewed in present work. Haas in 1.938 presented the expression:
μ=3·√2·π/α
Later in 1.951 Lenz noted that μ can be approximated with the formula:
μ=6·π5
In 1.990,I.J. Good,a British mathematician assembled eight conjectures of numerology for the ratio of the rest masses of the proton and the electron. Recently the professional approach to mathematically decode μ was done by Simon Plouffe. He used a large database of mathematical constants and specialized programs to directly find an expression. Alone with his main remarkable result for the expression for µ via Fibonacci and Lucas numbers and golden ratio he also noted that expression for µ using π can be improved as:
μ=6·π5+328/π8
4. Exact mathematical expression using Fibonacci and Lucas numbers
Simon Plouffe in his work «A search for a mathematical expression for mass ratios using a large database» proposed the mathematical expression:
µ=F10·F53/2·L515/32/φ1/16
So:
µ=55·53/2·1115/32/φ1/16
With numerical value μ=1.836,15267481714……. and Relative Standard Uncertainty 1×10-9. We propose the exact mathematical expression for the Proton to Electron Mass Ratio:
µ=1147/32·55/2·9.3495/76/φ21/16 (1)
With exact numerical value μ=1.836,15267343. However:
(2·φ-1)2=5 , φ5-φ-5=11 , (φ19-φ-19)=9.349
So:
µ32=(φ5-φ-5)47·(2·φ-1)160·(φ19-φ-19)40/19/φ42 (2)
Also:
µ32=φ-42·F5160·L547·L1940/19 (3)
The formula has an exact value,a symmetrical shape and a greater physical meaning than all types. It seems to be the formula of the Universe.
5. Exact mathematical expressions between mathematical constants
Exact arithmetic mathematical expressions for the Proton to Electron Mass Ratio μ:
1) µ=165∙[(ln10)11/7)1/3]
So:
7∙µ3=(5∙13)3∙[ln(2∙5)]11 (4)
2) µ=1.836+[2∙√77-11)-1 (5)
3) µ=210+29+28+25+23+22+2-3+2-6+2-7+2-8+2-12+2-14+2-16+2-17+2-20+2-21+2-23+2-24+2-27 (6)
4) µ=2∙54+4∙53+3∙52+2∙51+4∙5-1+8∙5-2+4∙5-3+2∙5-5+2∙5-7+3∙5-8+5-10+2∙5-11+2∙5-12 (7)
5) µ=1.836+(25∙3∙29/5∙7∙521) (8)
Exact mathematical expressions between the Proton to Electron Mass Ratio μ and Archimedes's constant π:
1) μ=64·π3-48·π+8·π-1+2·π-7+8·π-9+π-11+6·π-15+π-17 (9)
2) μ=(826·π)-(4.610/π)-(809/3) (10)
3) μ=6·π5+π-3+2·π-6+2·π-8+2·π-10+2·π-13+π-15 (11)
Exact mathematical expression between the Proton to Electron Mass Ratio μ and golden radio φ:
µ=φ15+φ12+φ10+2·φ5+φ3+φ-1+φ-3+φ-7+φ-12+φ-15+φ-17+φ-26+φ-31+φ-34 (12)
Exact mathematical expression between the Proton to Electron Mass Ratio μ and Euler's number e:
µ=e7+e6+2·e5+e3+2·e2+e1+4·e-1+e-2+e-3+e-4+e-5+2·e-8+2·e-10+e-11+2·e-16 (13)
Exact mathematical expression between the Proton to Electron Mass Ratio μ,Archimedes's constant π and golden radio φ:
μ=[π9-(3.981/40)]/10∙φ (14)
Exact mathematical expression between the Proton to Electron Mass Ratio μ,golden radio φ and Euler's number e:
µ=477·φ-74·e+446·√2+447·√3-108 (15)
Exact mathematical expression between the Proton to Electron Mass Ratio μ,Archimedes's constant π and Euler's number e:
µ=3·(eπ)2+9·(eπ)+495·(eπ)-1+11·(eπ)-2+11·(eπ)-4+22·(eπ)-5+11·(eπ)-6+8·(eπ)-7 (16)
Exact mathematical expression between the Proton to Electron Mass Ratio μ,Archimedes's constant π,golden radio φ and Euler's number e:
µ=544∙π+493∙φ-463∙e+588 (17)
6. Exact mathematical expressions between the Proton to Electron Mass Ratio,the Fine Structure constant and mathematical constants
Exact mathematical expressions between the Proton to Electron Mass Ratio μ,the Fine Structure constant α and mathematical constants:
1) μ=398·α+528/7·π+8 (18)
2) μ=(5/7)·(1.224·α+1.287·π-917·φ) (19)
3) μ=(1/18)∙(238∙α-1+10∙φ+400) (20)
4) µ=6·α-1+360·φ-165·π+345·e+12 (21)
5) µ=182·α+141·φ+495·π-66·e+231 (22)
6) µ=807·α+1.205·π-518·φ-411·e (23)
7) µ=15·α-1-3·Α+9·S-11·K-28·π-23·φ+e-30 (24)
8) μ=(1/5)⋅(69⋅α-1+52⋅QΑ+46⋅π-72⋅φ-46⋅π-111⋅e-27) (25)
9) μ=14⋅α-1+10⋅QΑ+4⋅A-5⋅S-K-17⋅φ-12⋅π-3 (26)
Κ the polygon circumscribing constant with value K=8,7000366252......
S the silver constant with value S=2+2∙cos(2∙π/7)=3,246979603717.........
A the Golden Apex with value A=eπ-7∙π-1=0,14954405765........
QΑ the Aristotle’s Quintessence with value QΑ=1,0191134319.........
7. Conclusions
All 26 mathematical expressions of the Proton to Electron Mass Ratio have exact value μ=1.836,15267343. They have been analysed in terms of their simplicity and numerical significance. We propose these mathematical expressions:
µ32=φ-42·F5160·L547·L1940/19
µ32=(φ5-φ-5)47·(2·φ-1)160·(φ19-φ-19)40/19/φ42
7∙µ3=(5∙13)3∙[(ln2∙5)11
μ=6·π5+π-3+2·π-6+2·π-8+2·π-10+2·π-13+π-15
µ=φ15+φ12+φ10+2·φ5+φ3+φ-1+φ-3+φ-7+φ-12+φ-15+φ-17+φ-26+φ-31+φ-34
µ=6·α-1+360·φ-165·π+345·e+12
µ=807·α+1.205·π-518·φ-411·e
µ=15·α-1-3·Α+9·S-11·K-28·π-23·φ+e-30
μ=14⋅α-1+10⋅QΑ+4⋅A-5⋅S-K-17⋅φ-12⋅π-3
References
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[3] www.math.stackexchange.com/
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https://vixra.org/abs/2110.0043, 2.021
[8] Stergios Pellis Exact Formula for the Fine Structure Constant α in Terms of the Golden Ratio φ
https://vixra.org/abs/2110.0053, 2.021
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viXra:1409.0099, 2.014
[10] Alexander Kritov A brief essay on numerology of the mass ratio of proton to electron
https://vixra.org/pdf/1410.0105v1.pdf, 2.014
[11] Michael A. Sherbon Quintessential Nature of the Fine-Structure Constant
https://vixra.org/pdf/1709.0083v1.pdf, 2.015
