A note on the diophantine equation $x^2+b^Y=c^z$

Le, Maohua

Geodesic

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A note on the diophantine equation

x^{2} + b^{Y} = c^{z}

Le, Maohua

Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1109-1116.

Voir la notice de l'article dans Czech Digital Mathematics Library

Résumé

Let

a

b

c

r

be positive integers such that

a^{2} + b^{2} = c^{r}

min (a, b, c, r) > 1

gcd (a, b) = 1, a

is even and

r

is odd. In this paper we prove that if

b \equiv 3 (\@ m o d 4)

and either

b

c

is an odd prime power, then the equation

x^{2} + b^{y} = c^{z}

has only the positive integer solution

(x, y, z) = (a, 2, r)

with

min (y, z) > 1

MR Zbl

Classification : 11D61
Mots-clés : exponential diophantine equation; Lucas number; positive divisor

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     author = {Le, Maohua},
     title = {A note on the diophantine equation $x^2+b^Y=c^z$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1109--1116},
     publisher = {mathdoc},
     volume = {56},
     number = {4},
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     mrnumber = {2280797},
     zbl = {1164.11319},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/CMJ_2006__56_4_a2/}
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Le, Maohua. A note on the diophantine equation $x^2+b^Y=c^z$. Czechoslovak Mathematical Journal, Tome 56 (2006) no. 4, pp. 1109-1116. https://geodesic-test.mathdoc.fr/item/CMJ_2006__56_4_a2/