A note on the regularity of the Diophantine pair $\protect \lbrace k,4k\pm 4\protect \rbrace $

Bo He; Keli Pu; Rulin Shen; Alain Togbé

doi:10.5802/jtnb.1055

A note on the regularity of the Diophantine pair

{k, 4 k \pm 4}

Bo He ^1,²; Keli Pu ³; Rulin Shen ¹; Alain Togbé 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

Journal de théorie des nombres de Bordeaux, Volume 30 (2018) no. 3, pp. 879-892.

Abstract
Résumé

Let $ε \in {\pm 1}$ and let $k$ be an integer such that $k \geq 2$ if $ε = - 1$ and $k \geq 1$ if $ε = 1$ . For positive integer $d$ , we prove that if the product of any two distinct elements of the set

{k, 4 k + 4 ε, 144 k^{3} + 240 k^{2} ε + 124 k + 20 ε, d}

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

d = 2304 k^{5} + 6144 k^{4} ε + 6112 k^{3} + 2784 k^{2} ε + 569 k + 42 ε .

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

Soit $ε \in {\pm 1}$ et soit $k$ un entier tel que $k \geq 2$ si $ε = - 1$ et $k \geq 1$ si $ε = 1$ . Pour tout eniter positif $d$ , nous démontrons que si le produit de deux éléments distincts de l’ensemble

{k, 4 k + 4 ε, 144 k^{3} + 240 k^{2} ε + 124 k + 20 ε, d}

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

d = 2304 k^{5} + 6144 k^{4} ε + 6112 k^{3} + 2784 k^{2} ε + 569 k + 42 ε .

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 ${k, 4 k + 4 ε, c, d}$ sont réguliers.

Received: 2017-06-08
Revised: 2017-10-01
Accepted: 2017-10-02
Published online: 2019-03-28

Zbl

DOI: 10.5802/jtnb.1055

Classification: 11D09, 11B37, 11J68, 11J86, 11Y65
Keywords: Diophantine $m$-tuples, Pell equations, Baker’s method, Reduction method

Author's affiliations:

Bo He ^{1, 2}; Keli Pu ³; Rulin Shen ¹; Alain Togbé 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

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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},
     zbl = {1446.11050},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/}
}

TY  - JOUR
AU  - Bo He
AU  - Keli Pu
AU  - Rulin Shen
AU  - Alain Togbé
TI  - A note on the regularity of the Diophantine pair $\protect \lbrace k,4k\pm 4\protect \rbrace $
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2018
SP  - 879
EP  - 892
VL  - 30
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/
DO  - 10.5802/jtnb.1055
LA  - en
ID  - JTNB_2018__30_3_879_0
ER  -

%0 Journal Article
%A Bo He
%A Keli Pu
%A Rulin Shen
%A Alain Togbé
%T A note on the regularity of the Diophantine pair $\protect \lbrace k,4k\pm 4\protect \rbrace $
%J Journal de théorie des nombres de Bordeaux
%D 2018
%P 879-892
%V 30
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/
%R 10.5802/jtnb.1055
%G en
%F JTNB_2018__30_3_879_0

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, Volume 30 (2018) no. 3, pp. 879-892. doi : 10.5802/jtnb.1055. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1055/

References
Cited by

[1] Joseph Arkin; V. E. jun. Hoggatt; Ernst G. Straus On Euler’s solution of a problem of Diophantus, Fibonacci Q., Volume 17 (1979), pp. 333-339 | MR | Zbl

[2] Alan Baker; Harold Davenport The equations $3 x^{2} - 2 = y^{2}$ and $8 x^{2} - 7 = z^{2}$ , Q. J. Math, Volume 20 (1969), pp. 129-137 | MR | Zbl

[3] Yann Bugeaud; Andrej Dujella; Maurice Mignotte On the family of Diophantine triples ${k - 1, k + 1, 16 k^{3} - 4 k}$ , Glasg. Math. J., Volume 49 (2007), pp. 333-344 | DOI | MR | Zbl

[4] Andrej Dujella The problem of the extension of a parametric family of Diophantine triples, Publ. Math., Volume 51 (1997) no. 3-4, pp. 311-322 | MR | Zbl

[5] Andrej Dujella An absolute bound for the size of Diophantine $m$ -tuples, J. Number Theory, Volume 89 (2001) no. 1, pp. 126-150 | DOI | MR | Zbl

[6] Andrej Dujella There are only finitely many Diophantine quintuples, J. Reine Angew. Math., Volume 566 (2004), pp. 183-214 | MR | Zbl

[7] Andrej Dujella; Attila Pethő A generalization of a theorem of Baker and Davenport, Q. J. Math, Volume 49 (1998) no. 195, pp. 291-306 | DOI | MR | Zbl

[8] Alan Filipin; Yasutsugu Fujita; Alain Togbé The extendibility of Diophantine pairs I: the general case, Glas. Mat., III. Ser., Volume 49 (2014) no. 1, pp. 25-36 | DOI | MR | Zbl

[9] Alan Filipin; Yasutsugu Fujita; Alain Togbé The extendibility of Diophantine pairs II: Examples, J. Number Theory, Volume 145 (2014), pp. 604-631 | DOI | MR | Zbl

[10] Yasutsugu Fujita The extensibility of Diophantine pairs ${k - 1, k + 1}$ , J. Number Theory, Volume 128 (2008) no. 2, pp. 322-353 | DOI | MR | Zbl

[11] Bo He; Alain Togbé; Volker Ziegler There is no Diophantine Quintuple, Trans. Am. Math. Soc., Volume 371 (2019) no. 9, pp. 6665-6709 | DOI | MR | Zbl

[12] Michel Laurent Linear forms in two logarithms and interpolation determinants II, Acta Arith., Volume 133 (2008) no. 4, pp. 325-348 | DOI | MR | Zbl

Cited by Sources: