A note on the regularity of the Diophantine pair {k,4k±4}
Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 879-892.

Soit ε{±1} et soit k un entier tel que k2 si ε=-1 et k1 si ε=1. Pour tout eniter positif d, nous démontrons que si le produit de deux éléments distincts de l’ensemble

{k,4k+4ε,144k3+240k2ε+124k+20ε,d}

augmenté de 1 est un carré parfait, alors d=9k+6ε ou

d=2304k5+6144k4ε+6112k3+2784k2ε+569k+42ε.

Par conséquence, en combinant ce résultat avec un résultat récent de Filipin, Fujita et Togbé, nous provons que tous les quadruplets diophantiens de la forme {k,4k+4ε,c,d} sont réguliers.

Let ε{±1} and let k be an integer such that k2 if ε=-1 and k1 if ε=1. For positive integer d, we prove that if the product of any two distinct elements of the set

{k,4k+4ε,144k3+240k2ε+124k+20ε,d}

increased by 1 is a perfect square, then d=9k+6ε or

d=2304k5+6144k4ε+6112k3+2784k2ε+569k+42ε.

Consequently, combining this result with a recent result of Filipin, Fujita and Togbé, we show that all Diophantine quadruples of the form {k,4k+4ε,c,d} are regular.

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DOI : https://doi.org/10.5802/jtnb.1055
Classification : 11D09,  11B37,  11J68,  11J86,  11Y65
Mots clés : Diophantine m-tuples, Pell equations, Baker’s method, Reduction method
@article{JTNB_2018__30_3_879_0,
     author = {Bo He and Keli Pu and Rulin Shen and Alain Togb\'e},
     title = {A note on the regularity of the {Diophantine} pair $\protect \lbrace k,4k\pm 4\protect \rbrace $},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {879--892},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {30},
     number = {3},
     year = {2018},
     doi = {10.5802/jtnb.1055},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/}
}
Bo He; Keli Pu; Rulin Shen; Alain Togbé. A note on the regularity of the Diophantine pair $\protect \lbrace k,4k\pm 4\protect \rbrace $. Journal de Théorie des Nombres de Bordeaux, Tome 30 (2018) no. 3, pp. 879-892. doi : 10.5802/jtnb.1055. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/

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