## Tokyo Journal of Mathematics

### On the Polynomial Quadruples with the Property $D(-1;1)$

#### Abstract

In this paper we prove, under some assumptions, that every polynomial $D(-1)$-triple in $\mathbb{Z}[X]$ can only be extended to a polynomial $D(-1;1)$-quadruple in $\mathbb{Z}[X]$ by polynomials $d^{\pm}$. More precisely, if $\{a,b,c;d\}$ is a polynomial $D(-1;1)$-quadruple, then $$d=d^{\pm}=-(a+b+c)+2(abc\pm rst)\,,$$ where $r$, $s$ and $t$ are polynomials from $\mathbb{Z}[X]$ with positive leading coefficients that satisfy $ab-1=r^2$, $ac-1=s^2$ and $bc-1=t^2$.

#### Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 527-540.

Dates
First available in Project Euclid: 20 November 2017

https://projecteuclid.org/euclid.tjm/1511221570

Mathematical Reviews number (MathSciNet)
MR3908808

Zentralblatt MATH identifier
07053490

Subjects
Primary: 11D09: Quadratic and bilinear equations
Secondary: 11D45: Counting solutions of Diophantine equations

#### Citation

BLIZNAC TREBJEŠANIN, Marija; FILIPIN, Alan; JURASIĆ, Ana. On the Polynomial Quadruples with the Property $D(-1;1)$. Tokyo J. Math. 41 (2018), no. 2, 527--540. https://projecteuclid.org/euclid.tjm/1511221570

#### References

• A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566 (2004), 183–214.
• A. Dujella, Diophantine $m$-tuples, http://web.math.pmf.unizg.hr/~duje/dtuples.html.
• A. Dujella, A. Filipin and C. Fuchs, Effective solution of the $D(-1)$-quadruple conjecture, Acta Arith. 128.4 (2007), 319–338.
• A. Dujella and C. Fuchs, A polynomial variant of a problem of Diophantus and Euler, Rocky Mountain J. Math. 33 (2003), 797–811.
• A. Dujella and C. Fuchs, Complete solution of the polynomial version of a problem of Diophantus, J. Number Theory 106 (2004), 326–344.
• A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 35–52.
• A. Dujella, C. Fuchs and R. F. Tichy, Diophantine $m$-tuples for linear polynomials, Period. Math. Hungar. 45 (2002), 21–33.
• A. Dujella and A. Jurasic, On the size of sets in a polynomial variant of a problem of Diophantus, Int. J. Number Theory 6 (2010), 1449–1471.
• A. Filipin, On the polynomial parametric family of the sets with the property $D(-1;1)$, Bol. Soc. Mat. Mexicana 16 (2010), 1–8.
• Y. Fujita, The $D(1)$-extensions of $D(-1)$-triples $\{1,2,c\}$ and integer points on the attached elliptic curves, Acta Arith. 128 (2007), 349–375.
• Y. Fujita, The Hoggatt-Bergum conjecture on $D(-1)$-triples $\{F_{2k+1}, F_{2k+3}, F_{2k+5}\}$ and integer points on the attached elliptic curves, Rocky Mountain J. Math. 39 (2009), 1907–1932.
• B. He and A. Togbé, On the $D(-1)$-triple $\{1,k^2+1,k^2+2k+2\}$ and its unique $D(1)$-extension, J. Number Theory 131 (2011), 120–137.