On Hilbert’s solution of Waring’s problem |
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Authors: | Paul Pollack |
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Institution: | 1. Department of Mathematics, University of Illinois, Urbana, IL, USA
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Abstract: | In 1909, Hilbert proved that for each fixed k, there is a number g with the following property: Every integer N ≥ 0 has a representation in the form N = x
1
k
+ x
2
k
+ … + x
g
k
, where the x
i
are nonnegative integers. This resolved a conjecture of Edward Waring from 1770. Hilbert’s proof is somewhat unsatisfying,
in that no method is given for finding a value of g corresponding to a given k. In his doctoral thesis, Rieger showed that by a suitable modification of Hilbert’s proof, one can give explicit bounds on
the least permissible value of g. We show how to modify Rieger’s argument, using ideas of F. Dress, to obtain a better explicit bound. While far stronger
bounds are available from the powerful Hardy-Littlewood circle method, it seems of some methodological interest to examine
how far elementary techniques of this nature can be pushed. |
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Keywords: | |
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