Continued fractions and the Thomson problem

We introduce new analytical approximations of the minimum electrostatic energy configuration of n electrons, E(n), when they are constrained to be on the surface of a unit sphere. Using 453 putative optimal configurations, we searched for approximations of the form E(n) = (n2/2)eg(n) where g(n) was obtained via a memetic algorithm that searched for truncated analytic continued fractions finally obtaining one with Mean Squared Error equal to 5.5549 x 10−8 for the model of the normalized energy (En(n)≡eg(n)≡2E(n)/n2). Using the Online Encyclopedia of Integer Sequences, we searched over 350,000 sequences and, for small values of n, we identified a strong correlation of the highest residual of our best approximations with the sequence of integers n defined by the condition that n2+12 is a prime. We also observed an interesting correlation with the behavior of the smallest angle α(n), measured in radians, subtended by the vectors associated with the nearest pair of electrons in the optimal configuration. When using both n−−√ and α(n) as variables a very simple approximation formula for En(n) was obtained with MSE= 7.9963 x 10−8 and MSE= 73.2349 for E(n). When expanded as a power series in infinity, we observe that an unknown constant of an expansion as a function of n−1/2 of E(n) first proposed by Glasser and Every in 1992 as −1.1039, and later refined by Morris, Deaven and Ho as −1.104616 in 1996, may actually be very close to −1.10462553440167 when the assumed optima for n≤200 are used.