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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
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