Abstract
We prove that the Leech lattice is the unique densest lattice in R^24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R^24 can exceed the Leech lattice's density by a factor of more than 1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice in R^8.
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