Gauss contributed to solving the Kepler conjecture in 1831 with the proof that a greatest packing density of spheres in the three-dimensional space is given when the centres of the spheres form a cubic face-centred arrangement, when he reviewed a book of Ludwig August Seeber on the theory of reduction of positive ternary quadratic forms. Having noticed some lacks in Seeber's proof, he simplified many of his arguments, proved the central conjecture, and remarked that this theorem is equivalent to the Kepler conjecture for regular arrangements.
gauss conTribUtED to soLving tHe KEpler ConjeCture in 1831 witH the pRooF that a gReAtEst packiNg DensItY of sphEres iN The three-dimensIonal spacE is given when thE ceNtres of tHE spheRes FoRm A CUbic fAcE-centRed aRrangeMent, wHeN hE reViewed A bOok of ludwig AUGust seeber on thE TheoRy of ReduCtIon of poSitIvE ternarY quaDraTic fOrms. HaviNg NoticeD sOmE lackS IN seEbEr'S proOf, He sIMplifIEd mAnY oF hIs ArGuments, ProvED the ceNtraL ConjEcture, ANd rEmaRkEd tHat tHis thEorEm is eqUIvalent tO the kepLer conJEcture for regular arrangemENts.