Geodesic
On Waring's problem (elementary methods)
Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 149-175

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The paper deals with two elementary methods for solving Waring's problem on the representation of numbers as a sum of equal exponents in powers of natural numbers. The first method is an elementary version of the original Hilbert's proof, and the second one simplifies and makes more precise the elementary Linnik's proof based on the estimation of the number of solutions of a certain system of Diophantine equations. 
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     author = {Yu. V. Nesterenko},
     title = {On {Waring's} problem (elementary methods)},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {149--175},
     publisher = {mathdoc},
     volume = {322},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a10/}
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Yu. V. Nesterenko. On Waring's problem (elementary methods). Zapiski Nauchnykh Seminarov POMI, Proceedings on number theory, Tome 322 (2005), pp. 149-175. http://geodesic.mathdoc.fr/item/ZNSL_2005_322_a10/