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


11D04 Linear Diophantine equations
Full Text: DOI Numdam EuDML


[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.