A new conjecture concerning the Diophantine equation $x^2 + b^y = c^z$
Cao, Zhenfu ; Dong, Xiaolei ; Li, Zhong
Proc. Japan Acad. Ser. A Math. Sci., Tome 78 (2002) no. 10, p. 199-202 / Harvested from Project Euclid
Text on Project Euclid
PDF
Alt PDF
In this paper, using a recent result of Bilu, Hanrot and Voutier on primitive divisors, we prove that if $a = |V_r|$, $b = |U_r|$, $c = m^2 + 1$, and $b \equiv 3 \pmod{4}$ is a prime power, then the Diophantine equation $x^2 + b^y = c^z$ has only the positive integer solution $(x,y,z) = (a,2,r)$, where $r > 1$ is an odd integer, $m \in \mathbf{N}$ with $2 \mid m$ and the integers $U_r$, $V_r$ satisfy $(m + \sqrt{-1} )^r = V_r + U_r \sqrt{-1}$.
Publié le : 2002-12-14
Classification:  Exponential Diophantine equation,  Lucas sequence,  primitive divisor,  Gauss integer,  11D61 
@article{1148392271,
     author = {Cao, Zhenfu and Dong, Xiaolei and Li, Zhong},
     title = {A new conjecture concerning the Diophantine equation $x^2 + b^y = c^z$},
     journal = {Proc. Japan Acad. Ser. A Math. Sci.},
     volume = {78},
     number = {10},
     year = {2002},
     pages = { 199-202},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1148392271}
}

Cao, Zhenfu; Dong, Xiaolei; Li, Zhong. A new conjecture concerning the Diophantine equation $x^2 + b^y = c^z$. Proc. Japan Acad. Ser. A Math. Sci., Tome 78 (2002) no. 10, pp.  199-202. http://gdmltest.u-ga.fr/item/1148392271/