Xie, Tiantian; Yang, Hai; Xu, Qian The positive integer solutions of elliptic curves \({y^2} = qx ({x^2} - 256)\). (Chinese. English summary) Zbl 1474.11087 J. Qufu Norm. Univ., Nat. Sci. 47, No. 1, 47-51 (2021). 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 Keywords:congruence; positive integer solution; elliptic curve PDFBibTeX XMLCite \textit{T. Xie} et al., J. Qufu Norm. Univ., Nat. Sci. 47, No. 1, 47--51 (2021; Zbl 1474.11087)