On the sums $S_χ(m)$

J. C. Peral

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On the diophantine equation $(x^{m} + 1) (x^{n} + 1) = y ²$

Maohua Le (1997)

Acta Arithmetica

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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) $(x^{m} + 1) (x^{n} + 1) = y ²$ , 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....