May 5 2000
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GENERALIZATION OF THE CLASSICAL WARING PROBLEM
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G.PALL’s THEOREM (1933). By any s ³ 6, any natural number n ³ 1 is representable as a sum of exactly s squares of positive integers, except for the numbers 1,2,3,..., s-1, and the numbers of the form s + {1,2,4,5,7,10,13}.
G.PALL’s THEOREM 2. By s=5, any natural number n ³ 1 is representable as a sum of exactly 5 squares of positive integers, except for the numbers 1,2,3,4, 5+{1,2,4,5,7,10,13} and the number 33.
A.ZENKIN’s THEOREM (1979). For any s ³ 14, any natural number n ³ 1 is representable as a sum of exactly s squares of positive integers, except for the numbers 1,2,3,..., s-1, and the numbers of the form s + Z(1,3), where the set, Z(1,3), is known explicitly.