Geodesic
On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
Acta Arithmetica, Tome 82 (1997) no. 1, pp. 17-26 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

Voir la notice de l'article

1. Introduction. Let ℤ, ℕ, ℚ be the sets of integers, positive integers and rational numbers respectively. In [7], Ribenboim proved that the equation    (1) $(x^m + 1)(x^n + 1) = y²$, x,y,m,n ∈ ℕ, x > 1, n > m ≥ 1, has no solution (x,y,m,n) with 2|x and (1) has only finitely many solutions (x,y,m,n) with 2∤x. Moreover, all solutions of (1) with 2∤x satisfy max(x,m,n) C, where C is an effectively computable constant. In this paper we completely determine all solutions of (1) as follows.   Theorem. Equation (1) has only the solution (x,y,m,n)=(7,20,1,2).
DOI : 10.4064/aa-82-1-17-26

Maohua Le 1

1 
@article{10_4064_aa_82_1_17_26,
     author = {Maohua Le},
     title = {On the diophantine equation $(x^m + 1)(x^n + 1) = y{\texttwosuperior}$},
     journal = {Acta Arithmetica},
     pages = {17--26},
     year = {1997},
     volume = {82},
     number = {1},
     doi = {10.4064/aa-82-1-17-26},
     language = {fr},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/aa-82-1-17-26/}
}
TY  - JOUR
AU  - Maohua Le
TI  - On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
JO  - Acta Arithmetica
PY  - 1997
SP  - 17
EP  - 26
VL  - 82
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4064/aa-82-1-17-26/
DO  - 10.4064/aa-82-1-17-26
LA  - fr
ID  - 10_4064_aa_82_1_17_26
ER  - 
%0 Journal Article
%A Maohua Le
%T On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
%J Acta Arithmetica
%D 1997
%P 17-26
%V 82
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4064/aa-82-1-17-26/
%R 10.4064/aa-82-1-17-26
%G fr
%F 10_4064_aa_82_1_17_26
Maohua Le. On the diophantine equation $(x^m + 1)(x^n + 1) = y²$. Acta Arithmetica, Tome 82 (1997) no. 1, pp. 17-26. doi: 10.4064/aa-82-1-17-26

Cité par Sources :