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The positive integer solutions of elliptic curves \({y^2} = qx ({x^2} - 256)\). (Chinese. English summary) Zbl 1474.11087

Summary: By using the properties of congruence and the methods of elementary number theory, it is proved that the elliptic curve \({y^2} = qx ({x^2} - 256)\) has other positive integer points except integer solutions \((0, 0)\) and \((16, 0)\), that is to say: (i) if \(q = 5\), then the elliptic curve in title has integer points \((x, y) = (20, 120), (144, 3840)\); (ii) if \(q = 29\), then it has integer point \((x, y) = (156816, 334414080)\); (iii) if \(q = 41\), then it has integer point \((x, y) = (25, 615)\); (iv) if \(q \ne 5, 29, 41\), then it has at most one positive integer solution \((x, y)\), where \(q\) is a positive odd number without squared factor.

MSC:

11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
11G07 Elliptic curves over local fields
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