## A Waring problem |

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2n_l American Journal Apply Lemma Arts British Columbia combine complete the proof consecutive Consider decreases degree DEPARTMENT difference Employing equation evaluates exceed exceptional exist express f values fewer values Finally follows Hence holds ideal inclusive are represented increase Induction inequality integral n-th powers interval p2q IVAN MORTON NIVEN Journal of Mathematics L. E. Dickson LARGER EXPONENTS largest latter least u satisfying LEMMA 9 Lemma l0 LEMMAS OF ASCENT lies log LQ maximum minimum namely obtain P2qu paper placed positive integer PRELIMINARY LEMMAS Professors prove Quarter remains replaced require respectively result satisfying 9 Section seven shown sum of 2n sum of q Take THEOREM theory true University of British University of Chicago Vancouver verified WARING PROBLEM Waring's problem whence write written yield