Geodesic
The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$
Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 375-383
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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.
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.
DOI : 10.1007/s10587-013-0023-3
Classification : 11D25, 11G05, 14G05
Keywords: elliptic curve; integral point; quadratic diophantine equation 
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Yang, Hai; Fu, Ruiqin. The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$. Czechoslovak Mathematical Journal, Tome 63 (2013) no. 2, pp. 375-383. doi: 10.1007/s10587-013-0023-3

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