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Sums of like powers and some dense sets. (English) Zbl 1224.11041
The authors introduce the notion of $$P$$-sequences. These are sequences defined recursively. First define $$(-1,1)$$ as a $$P$$-sequence. Next, if $$a_0=a_k$$ and $$(a_0,\ldots,a_k)$$ is a $$P$$-sequence, then $$(a_0,\dots,a_k,a_k,\dots,a_0)$$ is a $$P$$-sequence. If $$a_0=-a_k$$ and $$(a_0,\ldots,a_k)$$ is a $$P$$-sequence, then $$(a_0,\dots,a_{k-1},0,-a_{k-1},\dots,-a_0)$$ is a $$P$$-sequence. Finally each $$P$$-sequence can be obtained only by a finite use of the above rules. The authors study the properties of $$P$$-sequences and apply their properties to the study of representability of real numbers. Another application consists of finding how to generate the Prouhet–Tarry–Escott pairs.
##### MSC:
 11B83 Special sequences and polynomials 11D41 Higher degree equations; Fermat’s equation 11B05 Density, gaps, topology 11B75 Other combinatorial number theory 11Y50 Computer solution of Diophantine equations
##### Keywords:
$$P$$-sequences; dense-expandable sequences
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