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Integral points on the elliptic curve \(y^2 = x(x - p)( x - q)\). I. (Chinese. English summary) Zbl 1424.11075

Summary: Let \(p, q\) be odd primes and \(m\) be positive odd number with \(p + 2^m = q\), \(p \equiv 3\pmod 4\). In this paper, we prove that if \(m = 1\) or \(3\), then the elliptic curve \(y^2 = x(x - p)(x - q)\) \((x > q)\) has at most one integral point \((x, y)\); if \(m \geq 5\), then the elliptic curve has at most two integral points \((x, y)\). Additionally, all integral points of the elliptic curve when \((p, q) = (71, 103)\) are given.

MSC:

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