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The integral points on elliptic curves y 2 = x 3 + (36n 2 − 9)x − 2(36n 2 − 5)
Authors:Hai Yang  Ruiqin Fu
Institution:1. School of Science, Xi’an Polytechnic University, Xi’an, Shaanxi, 710048, P.R. China
2. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, 710062, P.R. China
3. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, 710062, P.R. China
4. School of Science, Xi’an Shiyou University, Xi’an, Shaanxi, 710065, P.R. China
Abstract:Let n be a positive odd integer. In this paper, combining some properties of quadratic and quartic diophantine equations with elementary analysis, we prove that if n > 1 and both 6n 2 ? 1 and 12n 2 + 1 are odd primes, then the general elliptic curve y 2 = x 3+(36n 2?9)x?2(36n 2?5) has only the integral point (x, y) = (2, 0). By this result we can get that the above elliptic curve has only the trivial integral point for n = 3, 13, 17 etc. Thus it can be seen that the elliptic curve y 2 = x 3 + 27x ? 62 really is an unusual elliptic curve which has large integral points.
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