A Numerical Curiosity of the Proton-Electron Mass Ratio

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A Numerical Curiosity of the Proton-Electron Mass Ratio. A Numerical Curiosity of the Proton-Electron Mass Ratio. Karl G. Kreuzer. 2018-08-09.
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... ∼

π = 1.570 796 327... where π = arccos(−1) = 3.1415... . 2

Hence, 23 π R ∼ 6 × 3 × 1 + 2 + × 10−4 3 2 2

3





and π ∼ 2 ×



23 R − 1 − 62 × 33 32



× 104 = 3.1414376... .

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 17 20260 π 2 3 −22 −4 R ∼ (2 × 3) × 3 × 1 + 2 + (1 − ) × × 10 + × × 10 = 36 × 27 × 3 20261 2 9 20261 2   20260 20260 −4 −4 ×π ×10 = 3 × 18 × 2 × 17 + 9 × × 10 × π = 1836.152 673... , = 2×3×17×18+18×27× 20261 20261 which agrees in the first 10 digits with the measured value of the proton-electron mass ratio. It should be noted that this curios expression is a highly constructed expression. One should not infer any profound relationship between π and 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=17 prime 20261 : https://primes.utm.edu/curios/page.php?short=20261

Karl G. Kreuzer

2

2018-08-09