Is there a polynomial D(2X + 1)-quadruple?

Zrinka Franušić; Ana Jurasić

doi:10.2478/auom-2025-0019

.blurhash-client-img { display: none !important; }

Is there a polynomial D(2X + 1)-quadruple?

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 33 (2025): Issue 2 (June 2025)

By: Zrinka Franušić and Ana Jurasić

Open Access

|Jun 2025

Abstract

In this paper, we show that there does not exist a polynomial D(2X+ 1)-quadruple {a, b, c, d}, such that 0 < a < b < c < d and deg d = deg b.

References

N. C. Bonciocat, M. Cipu and M. Mignotte, There is no Diophantine D(−1)-quadruple, J. London Math. Soc. 105 (2022), 63–99.
Search in Google Scholar Back to article
K. Chakraborty, S. Gupta and A. Hoque, On a conjecture of Franušić and Jadrijević: Counter-examples, Results Math. 78 (2023), Article 18.
Search in Google Scholar Back to article
Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers, (I. G. Bashmakova, Ed.) (Nauka 1974), 85–86, 215–217.
Search in Google Scholar Back to article
A. Dujella, Diophantine m-tuples, https://web.math.pmf.unizg.hr/~duje/dtuples.html
Search in Google Scholar Back to article
A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65 (1993), 15–27.
Search in Google Scholar Back to article
A. Dujella, Some polynomial formulas for Diophantine quadruples, Grazer Math. Ber. 328 (1996), 25–30.
Search in Google Scholar Back to article
A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32 (1997), 1–10.
Search in Google Scholar Back to article
A. Dujella and C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus, J. Number Theory 106 (2004), 326–344.
Search in Google Scholar Back to article
A. Dujella, C. Fuchs and R. F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002), 21–33.
Search in Google Scholar Back to article
A. Dujella, C. Fuchs and G. Walsh, Diophantine m-tuples for linear polynomials. II. Equal degrees, J. Number Theory, 120 (2006), 213–228.
Search in Google Scholar Back to article
A. Dujella and A. Jurasić, On the size of sets in a polynomial variant of a problem of Diophantus, Int. J. Number Theory, 6 (2010), 1449–1471.
Search in Google Scholar Back to article
A. Dujella and A. Jurasić, Some Diophantine triples and quadruples for quadratic polynomials, J. Comb. Number Theory 3(2) (2011), 123141.
Search in Google Scholar Back to article
A. Dujella and F. Luca, On a problem of Diophantus with polynomials, Rocky Mountain J. Math., 37 (2007), 131–157.
Search in Google Scholar Back to article
A. Filipin and A. Jurasić, A polynomial variant of a problem of Diophantus and its consequences, Glas. Mat. Ser. III, 54 (2019), 21–52.
Search in Google Scholar Back to article
A. Filipin and A. Jurasić, On the size of Diophantine m-tuples for linear polynomials, Miskolc Math. Notes, 17 (2016), 2; 861–876.
Search in Google Scholar Back to article
[16] Z. Franušić, A Diophantine problem in ℤ[ (1+d)/2 ] \mathbb{Z}\left[ {\left( {1 + \sqrt d } \right)/2} \right] , Studia Sci. Math. Hungar. 46 (2009), 103–112.
Back to article
[17] Z. Franušić, Diophantine quadruples in the ring of integers of ℚ(23) \mathbb{Q}\left( {\root 3 \of 2 } \right) , Miskolc Math. Notes 14 (2013), 893–903.
Back to article
[18] Z. Franušić, Diophantine quadruples in the ring ℤ[ 2 ] \mathbb{Z}\left[ {\sqrt 2 } \right] , Math. Commun. 9 (2004), 141–148.
Back to article
[19] Z. Franušić, Diophantine quadruples in ℤ[ 4k+3 ] \mathbb{Z}\left[ {\sqrt {4k + 3} } \right] , Ramanujan J. 17 (2008), 77–88.
Back to article
[20] Z. Franušić and B. Jadrijević, D(n)-quadruples in the ring of integers of ℚ(2,3) \mathbb{Q}\left( {\sqrt 2 ,\sqrt 3 } \right) , Math. Slovaca 69 (2019), 1263–1278.
Back to article
Z. Franušić and A. Jurasić, On nonexistence of D(n)-quadruples, Math. Slovaca 74 (2024), No. 4, 835–844
Search in Google Scholar Back to article
[22] Z. Franušić and I. Soldo, The problem of Diophantus for integers of ℚ(-3) \mathbb{Q}\left( {\sqrt { - 3} } \right) , Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 18 (2014), 15–25.
Back to article
B. W. Jones, A second variation of a problem of Davenport and Diophantus, Fibonacci Quart. 15 (1977), 323–330.
Search in Google Scholar Back to article
B. W. Jones, A variation of a problem of Davenport and Diophantus, Quart. J. Math. Oxford Ser. (2) 27 (1976), 349–353.
Search in Google Scholar Back to article
LJ. Jukić Matić, On D(w)-quadruples in the rings of integers of certain pure number fields, Glas. Mat. Ser. III 49 (2014), 37–46.
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2025-0019 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

Journal RSS Feed

Language: English

Page range: 67 - 88

Submitted on: Oct 30, 2024

Accepted on: Mar 15, 2025

Published on: Jun 3, 2025

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

D(n)-quadruples,

difference of two squares,

ring of polynomials

Related subjects:

Mathematics,

General mathematics

© 2025 Zrinka Franušić, Ana Jurasić, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 33 (2025): Issue 2 (June 2025)