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11 represents attended biquadrates coefficients College concludes corollaries Corresponding derived from f Determination Dickson displayed dissertation double number employed equation example excluded express Extensions of Waring's fifth following Theorem form derived form of weight form with Sf forms of order fourth Further given h 3b Hence implies Impossible LIBRARIES MARGARET EVELYN MAUCH Math minimum nine cubes Notations and Definitions oases obtain order 11 paper partition of f positive integers possible Professor Proof proved received the degree repre represent all integers represent all positive represent every positive represent the simple schools Science sents all positive sets of values seventh powers Similarly simple number sixth powers South Dakota table gives TABLE II CONTINUED Theorem 7.5 University of Chicago verified to represent Waring Waring's theorem weight nine Write Yang's