Aug 9, 2018 - The proton-electron mass ratio R = 1836.152 673 89 up to decimal places is 1836, which is the product of prime 17 and 108, which both can be ...
A Numerical Curiosity of the Proton-Electron Mass Ratio
A Numerical Curiosity of the Proton-Electron Mass Ratio Karl G. Kreuzer 2018-08-09
The proton-electron mass ratio R = 1836.152 673 89 up to decimal places is 1836, which is the product of prime 17 and 108, which both can be expressed by numbers 2 and 3 23 23 8 3 2 2 3 2 5 2 3 2 3 1836 = 17 × 108 = (2 + 3 ) × (2 × 3 ) = 2 × 3 × 1 + 2 = 6 × 3 × 1 + 2 = 6 × 3 × 1 + . 3 3 9 Including the decimal places of R , leads to the expression 8 8 2 3 −4 2 3 R = 6 × 3 × 1 + + 0.000 157 070 991... = 6 × 3 × 1 + + 1.570 709 91... × 10 9 9 Obviously, 1.570 709 91... ∼
Now, as an ansatz, let 1 π 23 −4 2 3 , R = 6 × 3 × 1 + 2 + (1 − ) × × 10 3 a 2
Karl G. Kreuzer
1
2018-08-09
A Numerical Curiosity of the Proton-Electron Mass Ratio
which determines the factor 1 −
1 a
and a , respectively. Using R = 1836.152 673 89 , leads to
a = 20261.334 34... . Note that 20261 is the smallest five-digit prime ABCDE for which the all of the following are also prime: ABCD + E , ABC + DE , AB + CDE , A + BCDE , AB + CD + E , A + B + C + D + E . Finally, neglecting the decimal places of a, an approximate algebraic representation of R in terms of π and primes 2 and 3 and 20261 is obtained 23 1 π 2 3 −22 R ∼ (2 × 3) × 3 × 1 + 2 + (1 − ) × × 10 = 1836.152 673... , 3 20261 2 which agrees in the first 10 digits with the measured value of the proton-electron mass ratio.
References : proton-electron mass ratio: https://physics.nist.gov/cgi-bin/cuu/Value?mpsme prime 17 : https://primes.utm.edu/curios/page.php?short=20261 prime 20261 : https://primes.utm.edu/curios/page.php?short=20261
