# mathematical modeling – How can one decide between structure or coincidence regarding the fine structure constant?

This question originates the following observations about the fine structure constant, $$alpha$$.

Measurements of $$alpha^{-1}$$ yield values slightly smaller than $$137.036$$, and a good rational fit for all of them is $$frac{34259}{250}$$. The choice of $$250$$ does not seem to be decimally biased, because one can multiply any value close to $$137.035999$$ for all integers up to say $$300$$ and $$250$$ emerges as a proper denominator more clearly than $$113$$ for $$pi$$.

One of the alternative approximate expressions for $$alpha^{-1}$$ is $$5^3+frac{5^2}{2}-frac{1}{2}+5^{-2}-frac{5^{-3}}{2}$$ After some easy manipulation, this can be written as $$frac{1}{2^3}(10^3+2^3ยท3(1+frac{3}{10^3}))$$

I would like to learn how to decide between structure or coincidence, based on data. For instance, how should one compare the complexities of the approximate value of the constant ($$137.036$$) and the above expression that suggests some internal structure?

An equivalent way to see $$137$$ (only the integer part of $$alpha^{-1}$$) is the following image, corresponding to the polynomial $$(4n+1)^3+3(2n)^2$$ at $$n=1$$.

I would classify this as a figurate number, as it is the sum of related (by $$n$$) figurate numbers (even squares and related cubes). But how can the structureness of this representation be evaluated?