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A recursion on quadruples. (English) Zbl 0543.10004

Let \(z=(a,b,c,d)\) be a quadruple of real numbers and let \(P(z)=(| a- b|,| b-c|,| c-d|,| d-a|).\) It is known that \(P^ n(z)=(0,0,0,0)=0\) for some integer n if a,b,c,d are rational integers and \(P^ n(z)\) is defined by \(P^ 1(z)=P(z)\) and \(P^ m(z)=P(P^{m-1}(z)).\) If a,b,c,d are real numbers then there are quadruples z such that \(P^ n(z)\neq 0\) for any n; e.g. if \(\lambda\) is the real root of the polynomial \(x^ 3+2x^ 2-2\) and \(k>0\) is a real number, then \(P^ n(kz)=\lambda^ nkz\neq 0\) for \(z=(1,1+\lambda,(1+\lambda)^ 2,(1+\lambda)^ 3).\) The author proves that these are essentially all the quadruples of real numbers for which \(P^ n(z)\neq 0\) for any n. The paper is an English version of an earlier one written in Hungarian [Mat. Lapok 29, 345-347 (1981; Zbl 0507.10006)].

MSC:

11A63 Radix representation; digital problems
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