The previous post mentioned that Avogadro’s constant is approximately 24!. Are there other physical constants that are nearly factorials?
I searched SciPy’s collection of physical constants looking for values that are either nearly factorials or nearly reciprocals of factorials.
The best example is the “classical electron radius” re which is 2.818 × 10-15 m and 1/17! = 2.811 × 10-15.
Also, the “Hartree-Hertz relationship” Eh/h equals 6.58 × 1015 and 18! = 6.4 × 1015. (Eh is the Hartree energy and h is Plank’s constant.)
Here’s the Python code I used to discover these relationships.
from scipy.special import gammaln from math import log, factorial from scipy.optimize import brenth from scipy.constants import codata def inverse_factorial(x): # Find r such that gammaln(r) = log(x) # So gamma(r) = x and (r-1)! = x r = brenth(lambda t: gammaln(t) - log(x), 1.0, 100.0) return r-1 d = codata.physical_constants for c in d: (value, unit, uncertainty) = d[ c ] x = value if x < 0: x = abs(x) if x < 1.0: x = 1.0/x r = inverse_factorial(x) n = round(r) # Use n > 6 to weed out uninteresting values. if abs(r - n) < 0.01 and n > 6: fact = factorial(n) if value < 1.0: fact = 1.0/fact print c, n, value, fact