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In 1611, Kepler 
proposed that close packing (cubic or hexagonal) is the densest possible Sphere Packing (has
the greatest 
), and this assertion is known as the Kepler conjecture. Finding the densest (not necessarily periodic)
packing of spheres is known as the Kepler Problem.
A putative proof of the Kepler conjecture was put forward by W.-Y. Hsiang (Hsiang 1992, Cipra 1993), but was subsequently determined to be flawed (Conway et al. 1994, Hales 1994). According to J. H. Conway, nobody who has read Hsiang's proof has any doubts about its validity: it is nonsense. In 1998, Hales completed a full proof of the conjecture which appears in a series of papers totaling more than 250 pages.
See also Dodecahedral Conjecture, Kepler Problem
References
Cipra, B. ``Gaps in a Sphere Packing Proof?'' Science 259, 895, 1993.
Conway, J. H.; Hales, T. C.; Muder, D. J.; and Sloane, N. J. A. ``On the Kepler Conjecture.''
Math. Intel. 16, 5, Spring 1994.
Eppstein, D. ``Sphere Packing and Kissing Numbers.''
http://www.ics.uci.edu/~eppstein/junkyard/spherepack.html.
Hales, T. C. ``The Sphere Packing Problem.'' J. Comput. Appl. Math. 44, 41-76, 1992.
Hales, T. C. ``Remarks on the Density of Sphere Packings in 3 Dimensions.'' Combinatori 13, 181-197, 1993.
Hales, T. C. ``The Status of the Kepler Conjecture.'' Math. Intel. 16, 47-58, Summer 1994.
Hales, T. C. `The Kepler Conjecture.''
http://www.math.lsa.umich.edu/~hales/countdown/.
Hsiang, W.-Y. ``On Soap Bubbles and Isoperimetric Regions in Noncompact Symmetrical Spaces. 1.''
Tôhoku Math. J. 44, 151-175, 1992.
Hsiang, W.-Y. ``A Rejoinder to Hales's Article.'' Math. Intel. 17, 35-42, Winter 1995.