| Title:
|
The integral points on elliptic curves $y^2=x^3+(36n^2 -9)x-2(36n^2-5)$ (English) |
| Author:
|
Yang, Hai |
| Author:
|
Fu, Ruiqin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
63 |
| Issue:
|
2 |
| Year:
|
2013 |
| Pages:
|
375-383 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
elliptic curve |
| Keyword:
|
integral point |
| Keyword:
|
quadratic diophantine equation |
| MSC:
|
11D25 |
| MSC:
|
11G05 |
| MSC:
|
14G05 |
| idZBL:
|
Zbl 06236417 |
| idMR:
|
MR3073964 |
| DOI:
|
10.1007/s10587-013-0023-3 |
| . |
| Date available:
|
2013-07-18T14:52:39Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143318 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| . |
