Magyar, Zoltan A recursion on quadruples. (English) Zbl 0543.10004 Am. Math. Mon. 91, 360-362 (1984). 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)]. Cited in 1 Document MSC: 11A63 Radix representation; digital problems Keywords:permutation; recursion; cyclic difference; quadruple of real numbers Citations:Zbl 0482.10009; Zbl 0507.10006 PDFBibTeX XMLCite \textit{Z. Magyar}, Am. Math. Mon. 91, 360--362 (1984; Zbl 0543.10004) Full Text: DOI