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A note on the diophantine equation k 2 - 1 = q n + 1

Maohua Le — 1998

Colloquium Mathematicae

In this note we prove that the equation k 2 - 1 = q n + 1 , q ≥ 2 , n ≥ 3 , has only finitely many positive integer solutions ( k , q , n ) . Moreover, all solutions ( k , q , n ) satisfy k 10 10 182 , q 10 10 165 and n 2 · 10 17 .

On the diophantine equation ( x m + 1 ) ( x n + 1 ) = y ²

Maohua Le — 1997

Acta Arithmetica

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 ) = y ² , 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)...

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