Power values of certain quadratic polynomials

Flatters, Anthony

Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 3, pp. 645-660.

In this article we compute the $q$th power values of the quadratic polynomials $f\in \mathbb{Z}\left[x\right]$ with negative squarefree discriminant such that $q$ is coprime to the class number of the splitting field of $f$ over $\mathbb{Q}$. The theory of unique factorisation and that of primitive divisors of integer sequences is used to deduce a bound on the values of $q$ which is small enough to allow the remaining cases to be easily checked. The results are used to determine all perfect power terms of certain polynomially generated integer sequences, including the Sylvester sequence.

Soit $f$ un polynôme quadratique à coefficients entiers avec discriminant sans carré parfait et $q>1$ un entier tel que $q$ et le nombre de classes du corps de rupture de $f$ sont premiers entre eux. Dans cet article, nous calculons les puissances q-ième qui apparaissent comme valeurs entières de $f$. La théorie des diviseurs primitifs de suites d’entiers permet de déduire une borne sur les valeurs possibles de $q$ qui est suffisamment petite pour que les cas restants puissent facilement être vérifiés. Ces résultats permettent de trouver toutes les puissances parfaites qui apparaissent dans certaines suites polynômiales récursives entières, y compris la suite de Sylvester.

Classification: 11B37, 11A41, 11B39
Keywords: Primitive divisor, Diophantine equation, Lucas sequence

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