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Primitive divisors of Lucas and Lehmer sequences. (English) Zbl 0832.11009

Let \(\alpha\), \(\beta\) be algebraic numbers such that \(\alpha+ \beta\) and \(\alpha \beta\) are relatively prime rational integers and \(\alpha/ \beta\) is not a root of unity. The sequence \((u_n )_{n\geq 0}\) defined by \(u_n= (\alpha^n- \beta^n)/ (\alpha- \beta)\) is called a Lucas sequence. If we replace the condition \(\alpha+ \beta\in \mathbb{Z}\) by \((\alpha+ \beta)^2\in \mathbb{Z}\) then, another sequence of rational integers can be defined by \(u_n= (\alpha^n- \beta^n)/ (\alpha- \beta)\) if \(n\) is odd and \(u_n= (\alpha^n- \beta^n)/ (\alpha^2- \beta^2)\) if \(n\) is even, which is called a Lehmer sequence. A prime number \(p\) is a primitive divisor of a Lucas (resp. Lehmer) number \(u_n\) if \(p\mid u_n \wedge p\nmid (\alpha- \beta)^2 u_2 \dots u_{n-1}\) (resp. \(p\nmid (\alpha^2- \beta^2)^2 u_3\dots u_{n-1})\).
C. L. Stewart has proved that any Lucas (resp. Lehmer) number \(u_n\) has a primitive divisor, provided that \(n> e^{452} 2^{67}\) (resp. \(n> e^{452} 4^{67}\)). He also proved that for any \(n= 5, 7, 8, 9, 10, 11, \dots\) (resp. \(n= 7, 9, 11, 13, 14, 15, \dots\)) there are at most finitely many Lucas (resp. Lehmer) numbers \(u_n\).
The author of the present paper explicitly determines all Lucas numbers \(u_n\) for \(n= 5, 7, 8, 9, 10, \dots, 30\) (such numbers actually occur only for \(n= 5, 7, 8, 10, 12, 13, 18\) and 30) and all Lehmer numbers \(u_n\) for \(n= 7, 9, 11, 13, 14, 15, \dots, 30\) (these actually occur only for \(n= 7, 9, 13, 14, 15, 18, 24, 26\) and 30). He reduces the problem of their determination to the explicit solution of diophantine equations (integer solutions) of type \(aX^2- bY^4 =c\) (\(a, b, c\) pairwise relatively prime integers with \(a\), \(b\) positive) in the cases \(n= 5, 8, 10\) and three instances of case \(n= 12\), and of Thue type for all other values of \(n\). These solutions of the non-trivial cases \(aX^2- bY^4 =c\) are due to Ljunggren, Cohn or Robbins, while the Thue equations (their degrees vary from 3 to 14) are solved by the general method of N. Tzanakis and B. M. M. de Weger [J. Number Theory 31, 99–132 (1989; Zbl 0657.10014)].
The paper is very readable, as the author gives sufficient technical details, which help the reader for a deeper understanding of the method. It is worth noticing that, to the best of the reviewer’s knowledge, this is the first paper in the literature in which Thue equations of this degree are solved.

MSC:

11B37 Recurrences
11D41 Higher degree equations; Fermat’s equation
11Y50 Computer solution of Diophantine equations

Citations:

Zbl 0657.10014
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References:

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