Minimal biquadratic energy of 5 particles on 2-sphere Alexander Tumanov 52A4052C35Discrete energyThomson\'s problemCauchy matrix Consider $n$ points on the unit 2-sphere. The potential energy of the interaction of two points is a function $f(r)$ of the distance $r$ between the points. The total energy $\mathcal{E}$ of $n$ points is the sum of the pairwise energies. The question is how to place the points on the sphere to minimize the energy $\mathcal{E}$. For the Coulomb potential $f(r)=1/r$, the problem goes back to Thomson (1904). The results for $n<5$ are simple and well known; we focus on the case $n=5$, which turns out to be difficult. Dragnev, Legg, and Townsend [P.\:D. Dragnev, D.\:A. Legg, and D.\:W. Townsend, \textit{Discrete logarithmic energy on the sphere}, Pacific J. Math. \textbf{207} (2002), no. 2, 345--358] give a solution of the problem for $f(r)= -\log r$ (known as Whyte's problem). Hou and Shao give a rigorous computer-aided solution for $f(r)= -r$, while Schwartz [R.\:E. Schwartz, \textit{The 5 electron case of Thomson's problem}, available at http://arxiv.org/abs/arXiv:1001.3702] gives one for Thomson's problem. Finally, we give a solution for biquadratic potentials. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5148 10.1512/iumj.2013.62.5148 en Indiana Univ. Math. J. 62 (2013) 1717 - 1731 state-of-the-art mathematics http://iumj.org/access/