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The kissing number k(n) is the maximal number of equal size nonoverlapping spheres in n dimensions that can touch another sphere of the same size. The number k(3) was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. Newton said that 12 should be the correct answer, while Gregory thought that 13 might be possible. This so-called "thirteen spheres problem" was finally solved by Schütte and van der Waerden in 1953 only.

Philippe Delsarte in 1973 described a "linear programming method" that does allow one to prove good bounds on k(n). Using this method in 1979 Levenshtein and at the same time Odlyzko and Sloane proved that k(8)=240 and k(24)=196560. The so-called 24-cell, a four-dimensional "Platonic solid", yields a configuration of 24 balls that would touch a given one in four dimensional space. It was proved (Arestov and Babenko) that Delsarte's method the best upper bound one could get is 25, i.e. k(4)=24 or 25. In this talk we present an extension of the Delsarte method and use it to prove that k(4)=24. We also present a new proof that k(3)=12.