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A lower bound for \(P(x^ 4+1)\). (English) Zbl 0633.10017

The author proves in detail that for \(x>3\), \(x^ 4+1\) is either prime or has a prime factor \(>73\). The method depends on considering solutions of certain Pell equations and gives an algorithm for finding all solutions in nonnegative integers \(x,\alpha_ 1,...,\alpha_ n\) of the equation \(x^ 2+1=p_ 1^{\alpha_ 1}p_ 2^{\alpha_ 2}...p_ n^{\alpha_ n}\) where \(p_ 1,...,p_ n\) are given prime numbers.
Reviewer: K.Ramachandra

MSC:

11D04 Linear Diophantine equations
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References:

[1] Borevitch, S.I.) and Schafarevitch, I.R.).- Théorie des Nombres.- Paris, Gauthier-Villars, 1967. · Zbl 0145.04901
[2] Cerlienco, L.), Mignotte, M.) and Piras, F.). - Suites Récurrentes Linéaires. Strasbourg, Publication de l’I.R.M.A., 1984. · Zbl 0626.10008
[3] Hardy, G.H.) and Wright, B.M.).- An Introduction of the Theory of Numbers. Oxford, Claredon Press, 1979. · Zbl 0423.10001
[4] Mignotte, M.).- P(x2 + 1) ≥ 17 si x ≥ 240. - C.R. Acad. Sc. t.301, series I, n°13, 1985. · Zbl 0591.10006
[5] Lucas, E.). - Théorie des Nombres. - Paris, Gauthier-Villars, 1891.
[6] Niven, I.) and Zuckerman, H.S.). - An Introduction to the Theory of Numbers. New York, John Wiley & Sons, 1960. · Zbl 0098.03602
[7] Pethö, A.) and De Weger, B.M.M.).- Products of prime Powers in Binary Recurrences Sequences, Mathematical Institute University of Leiden. The Netherlands, Report n.24, September 1985; Report n.29, November 1985.
[8] Størmer, C.).- Quelques Théorèmes sur l’équation de Pell x2 - Dy2 = ±1 et leurs applications,. - Vid.-Selsk. Skrifter. Math. Naturv. K1, 1897.
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