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

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1055
Classification: 11D09, 11B37, 11J68, 11J86, 11Y65
Keywords: Diophantine $m$-tuples, Pell equations, Baker’s method, Reduction method
He, Bo 1, 2; Pu, Keli 3; Shen, Rulin 1; Togbé, Alain 4

1 Department of Mathematics Hubei University for Nationalities Enshi, Hubei, 445000, P.R. China
2 Institute of Mathematics Aba Teachers University Wenchuan, Sichuan, 623000, P. R. China
3 Institute of Mathematics Aba Teachers University Wenchuan, Sichuan, 623000, P.R. China
4 Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S, U.S. 421 Westville IN 46391, USA
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He, Bo; Pu, Keli; Shen, Rulin; Togbé, Alain. A note on the regularity of the Diophantine pair $\protect \lbrace k,4k\pm 4\protect \rbrace $. Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 879-892. doi : 10.5802/jtnb.1055. http://www.numdam.org/articles/10.5802/jtnb.1055/

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