2006 Fiscal Year Final Research Report Summary
On solutions of polynomial Pell's equations and the continued fraction factorization algorithm
| Project/Area Number |
17540052
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Shibaura Institute of Technology (2006)
Hiroshima Institute of Technology (2005) |
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Principal Investigator |
YOKOTA Hisashi Shibaura Inst of Tech, Engineering, Professor, 工学部, 教授 (90210616)
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| Project Period (FY) |
2005 – 2006
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| Keywords | Polynomial Pell's equation / elliptic curve / continued fraction |
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| Research Abstract |
Study on the polynomial Pell's equation was first done by Abel in the connection of finding an elliptic integral which can be expressed using an elementary function. This was done in the field of rational numbers.
When we restrict solutions of the polynomial Pell's equation to be integer coefficient polynomial, the known result is only for a monic quartic polynomial. We have shown in 2003, a necessary and sufficient condition for the polynomial Pell's equation has a nontrivial integer coefficient polynomial solution for D = A^2+2C and A/C∈Q[x].
In this research, collaborating with Prof.Webb, we have studied the polynomial Pell's equation using the period of continued fraction expansions of √<D> in the connection with rational points on the elliptic curve arising from the partial quotients. We also have studied the polynomial Pell's equation by looking at the small periods.
For D a monic quartic polynomial, we are able to show that there is no period 3 continued fraction expansion.
For D a monic polynomial, we are able to show that the values of period of continued fraction expansions are even if and only if the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution.
For D a monic quartic polynomial, we are able to show that the polynomial Pell's equation X^2-DY^2 = 1 has a nontrivial solution in Q[x] if and only if the values of the period of continued fraction expansions are 2,4,6,8,10,14,18,22.
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Research Products
(4 results)
