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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.
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
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