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It will be demonstrated that there is a thin basis of order four of minimalcardinality #A(x) = O(x^.25). The current literature shows the existence of athin basis of order four of cardinality #A(x) = O(x^(.25+{\epsilon}),{\epsilon} > 0, but speculates on the existence of a thin basis of order fourof minimal cardinality.

Lagrange's Theorem On The Minimal Set Of Squares