Abstract By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation ( n k ) = ( m l ) + d for − 3 ≤ d ≤ 3 and ( k , l ) ∈ { ( 2 , 3 ) , ( 2 , 4 ) , ( 2 , 5 ) , ( 2 , 6 ) , ( 2 , 8 ) , ( 3 , 4 ) , ( 3 , 6 ) , ( 4 , 6 ) , ( 4 , 8 ) } . Moreover, we present some other observations of computational and theoretical nature concerning the title equation.

中文翻译：

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关于丢番图方程 (nk)=(ml)+d

摘要 通过找到某些椭圆曲线和超椭圆曲线上的所有积分点，我们完全求解丢番图方程( nk ) = ( ml ) + d for − 3 ≤ d ≤ 3 and ( k , l ) ∈ { ( 2 , 3 ) , ( 2 ) , 4), (2, 5), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8) }。此外，我们还提出了一些关于标题方程的计算和理论性质的其他观察结果。