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The Diophantine equation ( b n ) x + ( 2 n ) y = ( ( b + 2 ) n ) z

Min Tang, Quan-Hui Yang (2013)

Colloquium Mathematicae

Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation b x + 2 y = ( b + 2 ) z has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

The Diophantine equation D x ² + 2 2 m + 1 = y ⁿ

J. H. E. Cohn (2003)

Colloquium Mathematicae

It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.

The diophantine equation x 2 + 2 a · 17 b = y n

Su Gou, Tingting Wang (2012)

Czechoslovak Mathematical Journal

Let ℤ , ℕ be the sets of all integers and positive integers, respectively. Let p be a fixed odd prime. Recently, there have been many papers concerned with solutions ( x , y , n , a , b ) of the equation x 2 + 2 a p b = y n , x , y , n ∈ ℕ , gcd ( x , y ) = 1 , n ≥ 3 , a , b ∈ ℤ , a ≥ 0 , b ≥ 0 . And all solutions of it have been determined for the cases p = 3 , p = 5 , p = 11 and p = 13 . In this paper, we mainly concentrate on the case p = 3 , and using certain recent results on exponential diophantine equations including the famous Catalan equation, all solutions ( x , y , n , a , b ) of the equation x 2 + 2 a · 17 b = y n , x , y , n ∈ ℕ , gcd ( x , y ) = 1 , n ≥ 3 , a , b ∈ ℤ , a ≥ 0 , b ≥ 0 , are determined....

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