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

Maohua Le

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 4, page 1109-1116
  • ISSN: 0011-4642

Abstract

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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 ≡ 3 ( m o d 4 ) and either b or 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 .

How to cite

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Le, Maohua. "A note on the diophantine equation $x^2+b^Y=c^z$." Czechoslovak Mathematical Journal 56.4 (2006): 1109-1116. <http://eudml.org/doc/31093>.

@article{Le2006,
abstract = {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 $b\equiv 3\hspace\{4.44443pt\}(mod \; 4)$ and either $b$ or $c$ 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 $\min (y,z)>1$.},
author = {Le, Maohua},
journal = {Czechoslovak Mathematical Journal},
keywords = {exponential diophantine equation; Lucas number; positive divisor; exponential diophantine equation; Lucas number; positive divisor},
language = {eng},
number = {4},
pages = {1109-1116},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the diophantine equation $x^2+b^Y=c^z$},
url = {http://eudml.org/doc/31093},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Le, Maohua
TI - A note on the diophantine equation $x^2+b^Y=c^z$
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 4
SP - 1109
EP - 1116
AB - 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 $b\equiv 3\hspace{4.44443pt}(mod \; 4)$ and either $b$ or $c$ 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 $\min (y,z)>1$.
LA - eng
KW - exponential diophantine equation; Lucas number; positive divisor; exponential diophantine equation; Lucas number; positive divisor
UR - http://eudml.org/doc/31093
ER -

References

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  8. Diophantine Equations, Academic Press, London, 1969. (1969) Zbl0188.34503MR0249355
  9. Sur I’impossibilité de quelques equation á deux indéterminées, Norsk Matem. Forenings Skrifter 13 (1921), 65–82. (1921) 
  10. 10.4064/aa-63-4-351-358, Acta Arith. 63 (1993), 351–358. (1993) MR1218462DOI10.4064/aa-63-4-351-358
  11. 10.1090/S0025-5718-1995-1284673-6, Math. Comp. 64 (1995), 869–888. (1995) Zbl0832.11009MR1284673DOI10.1090/S0025-5718-1995-1284673-6

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