Geodesic
ON THE FAMILY OF DIOPHANTINE TRIPLES {k − 1, k + 1, 16k 3 − 4k}
Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 333-344

Voir la notice de l'article provenant de la source Cambridge University Press

It is proven that if k ≥ 2 is an integer and d is a positive integer such that the product of any two distinct elements of the set increased by 1 is a perfect square, then d = 4k or d = 64k 5−48k 3+8k. Together with a recent result of Fujita, this shows that all Diophantine quadruples of the form {k − 1, k + 1, c, d} are regular.
DOI : 10.1017/S0017089507003564
Mots-clés : 11D09, 11D25, 11J86, 11Y50 
BUGEAUD, YANN; DUJELLA, ANDREJ; MIGNOTTE, MAURICE. ON THE FAMILY OF DIOPHANTINE TRIPLES {k − 1, k + 1, 16k 3 − 4k}. Glasgow mathematical journal, Tome 49 (2007) no. 2, pp. 333-344. doi: 10.1017/S0017089507003564
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[1] 1. Baker, A. and Davenport, H., The equations 3x 2−2=y 2 and 8x 2−7=z 2 , Quart. J. Math. Oxford Ser. (2) 20 (1969), 129–137. Google Scholar | DOI

[2] 2. Baker, A. and Wüstholz, G., Logarithmic forms and group varieties, J. Reine Angew. Math. 442 (1993), 19–62. Google Scholar

[3] 3. Bennett, M. A., On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173–199. Google Scholar | DOI

[4] 4. Dickson, L. E., History of the theory of numbers, Vol. 2, Diophantinc analysis (Chelsea, New York, 1966), 513–520. Google Scholar

[5] 5. Diophantus of Alexandria, Arithmetics and the book of polygonal numbers (Bashmakova, I. G., Ed.), Nauka, Moscow, 1974 (in Russian), 103–104, 232. Google Scholar

[6] 6. Dujella, A., The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen 51 (1997), 311–322. Google Scholar | DOI

[7] 7. Dujella, A., An absolute bound for the size of Diophantine m-tuples, J. Number Theory 89 (2001), 126–150. Google Scholar | DOI

[8] 8. Dujella, A., There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183–214. Google Scholar

[9] 9. Dujella, A. and Pethö, A., A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291–306. Google Scholar | DOI

[10] 10. Gibbs, P., Some rational Diophantine sextuples, Glas. Mat. Ser. III 41 (2006), 195–203. Google Scholar | DOI

[11] 11. Gibbs, P., A generalised Stern-Brocot tree from regular Diophantine quadruples, preprint, math.NT/9903035. Google Scholar

[12] 12. Fujita, Y., The extensibility of Diophantine pairs k−1, k+1, preprint. Google Scholar

[13] 13. Heath, T. L., Diophantus of Alexandria: a study in the history of Greek algebra. Second edition. With a supplement containing an account of Fermat's theorems and problems connected with Diophantine analysis and some solutions of Diophantine problems by Euler (Powell's Bookstore, Chicago and Martino Publishing, Mansfield Center, 2003), 177–181. Google Scholar

[14] 14. Matveev, E. M., An explicit lower bound for a homogeneous rational linear form in logarithms of algebraic numbers. II (Russian), Izv. Ross. Akad. Nauk Ser. Mat. 64 (2000), no. 6, 125–180. Google Scholar

[15] 15. Mignotte, M., A kit on linear forms in three logarithms, preprint. Google Scholar

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