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Keywords:
elliptic curve; integral point; quadratic diophantine equation
Summary:
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.
References:
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