Geodesic
On the diophantine equation $(x^m + 1)(x^n + 1) = y²$
Acta Arithmetica, Tome 82 (1997) no. 1, pp. 17-26.

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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 
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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. http://geodesic.mathdoc.fr/articles/10.4064/aa-82-1-17-26/

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