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Author

Guillaume Hanrot

Bio: Guillaume Hanrot is an academic researcher from University of Bordeaux. The author has contributed to research in topics: Thue equation & Algebraic number field. The author has an hindex of 5, co-authored 5 publications receiving 529 citations.

Papers
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Journal ArticleDOI
TL;DR: In this paper, it was shown that every Lucas and Lehmer number without a primitive divisor has a primitive primitive for any value of n ≥ 30, where n is the number of nodes in the number.
Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

352 citations

Journal ArticleDOI
TL;DR: In this article, a general method for numerical solution of Thue equations is proposed, which allows one to solve in reasonable time Thue equation of high degree (provided necessary algebraic number theory data is available).

122 citations

Journal ArticleDOI
TL;DR: In this paper, the authors describe a method for complete solution of the superelliptic Diophantine equation with p = f(x) using linear forms in the logarithm.
Abstract: We describe a method for complete solution of the superelliptic Diophantine equation ay^p=f(x). The method is based on Baker's theory of linear forms in the logarithms. The characteristic feature of our approach (as compared with the classical method is that we reduce the equation directly to the linear forms in logarithms, without intermediate use of Thue and linear unit equations. We show that the reduction method of Baker and Davenport is applicable for superelliptic equations, and develop a very efficient method for enumerating the solutions below the reduced bound. The method requires computing the algebraic data in number fields of degree pn(n-1)/2 at most; in many cases this number can be reduced. Two examples with p=3 and n=4 are given.

37 citations

Journal ArticleDOI
TL;DR: It is shown that the knowledge of a subgroup of finite index is in fact sufficient and two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given.
Abstract: The main problem when solving a Thue equation is the computation of the unit group of a certain number field. It is shown that the knowledge of a subgroup of finite index is in fact sufficient. Two examples linked with the primitive divisor problem for Lucas and Lehmer sequences are given. || La principale difficulte lors de la resolution d'une equation de Thue reside dans le calcul du groupe des unites du corps de nombres associe. On montre qu'il suffit en fait de connaitre un sous-groupe d'indice fini de ce groupe. On donne deux exemples li

34 citations

Journal ArticleDOI
TL;DR: In this article, a method of resolution for the Thue equation was proposed, which takes advantage of the fact that the number field generated by a root of a root has small subfields.
Abstract: We consider the Thue equation $F(x,y)=a$, where $F$ is an irreducible form of degree $n\geq 3$.We describe a method of resolution which takes advantage of the fact that the number field generated by a root of $F(1,y)$ has small subfields. We illustrate this method by solving several real cyclotomic equations of degrees as large as 2505. || Considerons l'equation de Thue $F(x,y)=a$, avec $F$ une forme irreductible homogene de degre $n\geq 3$. Nous decrivons une methode de resolution permettant de tirer profit de l'existence de petits sous-corps du corps de nombres engendre par une racine

15 citations


Cited by
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Journal ArticleDOI
TL;DR: In this paper, it was shown that every Lucas and Lehmer number without a primitive divisor has a primitive primitive for any value of n ≥ 30, where n is the number of nodes in the number.
Abstract: We prove that for~${n>30}$, every~$n$-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

352 citations

Posted Content
TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem.
Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8, 144 and the only perfect powers in the Lucas sequence are 1, 4.

263 citations

Journal ArticleDOI
TL;DR: In this paper, the authors combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's last theorem.
Abstract: This is the first in a series of papers whereby we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we give new improved bounds for linear forms in three logarithms. We also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the Fibonacci sequence are 0, 1, 8 and 144 and the only perfect powers in the Lucas sequence are 1 and 4.

249 citations

Journal ArticleDOI
TL;DR: In this article, the authors developed techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms.
Abstract: In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2 , based upon the theory of Galois representations and modular forms. We subse- quently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan-Nagell type.

200 citations