Soit
In this article we compute the
Mots-clés : Primitive divisor; Diophantine equation; Lucas sequence
Anthony Flatters 1
@article{JTNB_2010__22_3_645_0, author = {Anthony Flatters}, title = {Power values of certain quadratic polynomials}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {645--660}, publisher = {Universit\'e Bordeaux 1}, volume = {22}, number = {3}, year = {2010}, doi = {10.5802/jtnb.737}, mrnumber = {2769336}, zbl = {1236.11018}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.737/} }
TY - JOUR AU - Anthony Flatters TI - Power values of certain quadratic polynomials JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 645 EP - 660 VL - 22 IS - 3 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.737/ DO - 10.5802/jtnb.737 LA - en ID - JTNB_2010__22_3_645_0 ER -
%0 Journal Article %A Anthony Flatters %T Power values of certain quadratic polynomials %J Journal de théorie des nombres de Bordeaux %D 2010 %P 645-660 %V 22 %N 3 %I Université Bordeaux 1 %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.737/ %R 10.5802/jtnb.737 %G en %F JTNB_2010__22_3_645_0
Anthony Flatters. Power values of certain quadratic polynomials. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 645-660. doi : 10.5802/jtnb.737. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.737/
[1] M. Abouzaid, Les nombres de Lucas et Lehmer sans diviseur primitif, J. Théor. Nombres Bordeaux, 18 (2006), pp. 299–313. | Numdam | MR | Zbl
[2] F. S. Abu Muriefah and Y. Bugeaud, The Diophantine equation
[3] S. A. Arif and F. S. Abu Muriefah, On the Diophantine equation
[4] S. A. Arif and A. S. Al-Ali, On the Diophantine equation
[5] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Cambridge Philos. Soc., 65 (1969), pp. 439–444. | MR | Zbl
[6] M. A. Bennett, N. Bruin, K. Győry, and L. Hajdu, Powers from products of consecutive terms in arithmetic progression, Proc. London Math. Soc. (3), 92 (2006), pp. 273–306. | MR | Zbl
[7] M. A. Bennett, K. Győry, M. Mignotte, and Á. Pintér, Binomial Thue equations and polynomial powers, Compos. Math., 142 (2006), pp. 1103–1121. | MR | Zbl
[8] A. Bérczes, B. Brindza, and L. Hajdu, On the power values of polynomials, Publ. Math. Debrecen, 53 (1998), pp. 375–381. | MR | Zbl
[9] Y. Bilu, On Le’s and Bugeaud’s papers about the equation
[10] Y. Bilu, G. Hanrot, and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math., 539 (2001), pp. 75–122. With an appendix by M. Mignotte. | MR | Zbl
[11] Y. F. Bilu and G. Hanrot, Solving superelliptic Diophantine equations by Baker’s method, Compositio Math., 112 (1998), pp. 273–312. | MR | Zbl
[12] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 (1997), pp. 235–265. Computational algebra and number theory (London, 1993). | MR | Zbl
[13] B. Brindza, On
[14] B. Brindza, J.-H. Evertse, and K. Győry, Bounds for the solutions of some Diophantine equations in terms of discriminants, J. Austral. Math. Soc. Ser. A, 51 (1991), pp. 8–26. | MR | Zbl
[15] Y. Bugeaud, Bounds for the solutions of superelliptic equations, Compositio Math., 107 (1997), pp. 187–219. | MR | Zbl
[16] Y. Bugeaud, On some exponential Diophantine equations, Monatsh. Math., 132 (2001), pp. 93–97. | MR | Zbl
[17] Y. Bugeaud, M. Mignotte, and S. Siksek, Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers, Ann. of Math. (2), 163 (2006), pp. 969–1018. | MR | Zbl
[18] H. Cohen, Pari-gp. www.parigp-home.de.
[19] J. H. E. Cohn, On square Fibonacci numbers, J. London Math. Soc., 39 (1964), pp. 537–540. | MR | Zbl
[20] A. Flatters, Arithmetic properties of recurrence sequences, PhD thesis, University of East Anglia, 2010.
[21] S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly, 70 (1963), pp. 403–405. | MR | Zbl
[22] K. Győry, L. Hajdu, and Á. Pintér, Perfect powers from products of consecutive terms in arithmetic progression, Compos. Math., 145 (2009), pp. 845–864. | MR | Zbl
[23] K. Győry, I. Pink, and A. Pintér, Power values of polynomials and binomial Thue-Mahler equations, Publ. Math. Debrecen, 65 (2004), pp. 341–362. | MR | Zbl
[24] K. Györy and Á. Pintér, Almost perfect powers in products of consecutive integers, Monatsh. Math., 145 (2005), pp. 19–33. | MR | Zbl
[25] K. Győry and Á. Pintér, On the resolution of equations
[26] K. Győry and Á. Pintér, Polynomial powers and a common generalization of binomial Thue-Mahler equations and
[27] V. Lebesgue, Sur l’impossibilité, en nombres entiers, de l’équation
[28] S. P. Mohanty, The number of primes is infinite, Fibonacci Quart., 16 (1978), pp. 381–384. | MR | Zbl
[29] A. Pethő, Full cubes in the Fibonacci sequence, Publ. Math. Debrecen, 30 (1983), pp. 117–127. | MR | Zbl
[30] D. Poulakis, Solutions entières de l’équation
[31] A. Schinzel and R. Tijdeman, On the equation
[32] T. N. Shorey and C. L. Stewart, On the Diophantine equation
[33] T. N. Shorey and C. L. Stewart, Pure powers in recurrence sequences and some related Diophantine equations, J. Number Theory, 27 (1987), pp. 324–352. | MR | Zbl
[34] T. N. Shorey and R. Tijdeman, Exponential Diophantine equations, vol. 87 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1986. | MR | Zbl
[35] C. L. Siegel, The integer solutions of the equation
[36] N. Sloane, Online encyclopedia of integer sequences. www.research.att.com/~njas/sequences.
[37] V. G. Sprindžuk, The arithmetic structure of integer polynomials and class numbers, Trudy Mat. Inst. Steklov., 143 (1977), pp. 152–174, 210. Analytic number theory, mathematical analysis and their applications (dedicated to I. M. Vinogradov on his 85th birthday). | MR | Zbl
[38] R. Tijdeman, Applications of the Gelʼfond-Baker method to rational number theory, in Topics in number theory (Proc. Colloq., Debrecen, 1974), North-Holland, Amsterdam, 1976, pp. 399–416. Colloq. Math. Soc. János Bolyai, Vol. 13. | MR | Zbl
[39] P. M. Voutier, An upper bound for the size of integral solutions to
Cité par Sources :