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Smale's 7-th problem concerns N-point configurations on the 2-dim spherewhich minimize the logarithmic pair-energy V_0(r) = -ln r averaged over thepairs in a configuration; here, r is the chordal distance between the pointsforming a pair. More generally, V_0(r) may be replaced by the standardizedRiesz pair-energy V_s(r)= (r^{-s} -1)/s, which becomes - ln r in the limit s to0, and the sphere may be replaced by other compact manifolds. This paperinquires into the concavity of the map from the integers N>1 into the minimalaverage standardized Riesz pair-energies v_s(N) of the N-point configurationson the 2-sphere for various real s. It is known that v_s(N) is strictlyincreasing for each real s, and for s<2 also bounded above, hence "overallconcave." It is (easily) proved that v_{-2}(N) is even locally strictlyconcave, and that so is v_s(2n) for s<-2. By analyzing computer-experimentaldata of putatively minimal average Riesz pair-energies v_s^x(N) for s in{-1,0,1,2,3} and N in {2,...,200}, it is found that {v}_{-1}^x(N) is locallystrictly concave, while v_s^x(N) is not always locally strictly concave for sin {0,1,2,3}: concavity defects occur whenever N in C^{x}_+(s) (an s-specificempirical set of integers). It is found that the empirical map C^{x}_+(s), withs in {-2,-1,0,1,2,3}, is set-theoretically increasing; moreover, the percentageof odd numbers in C^{x}_+(s), s in {0,1,2,3}, is found to increase with s. Theintegers in C^{x}_+(0) are few and far between, forming a curious sequence ofnumbers, reminiscent of the "magic numbers" in nuclear physics. It isconjectured that the "magic numbers" in Smale's 7-th problem are associatedwith optimally symmetric optimal-energy configurations.

"Magic" numbers in Smale's 7th problem