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1
Content available remote

The D(1)-extensions of D(-1)-triples 1,2,c and integer points on the attached elliptic curves

100%
Fujita Y.
Acta Arithmetica
|
2007
|
tom 128
|
nr 4
349-375
2
Content available remote

Torsion subgroups of elliptic curves with non-cyclic torsion over ℚ in elementary abelian 2-extensions of ℚ

100%
Fujita Y.
Acta Arithmetica
|
2004
|
tom 115
|
nr 1
29-45
3
Content available remote

Jeśmanowicz' conjecture with congruence relations

64%
Fujita Y., Miyazaki T.
Colloquium Mathematicum
|
2012
|
tom 128
|
nr 2
211-222
EN
Let a,b and c be relatively prime positive integers such that a²+b² = c². We prove that if $b ≡ 0 (mod 2^{r})$ and $b ≡ ±2^{r} (mod a)$ for some non-negative integer r, then the Diophantine equation $a^{x} + b^{y} = c^z$ has only the positive solution (x,y,z) = (2,2,2). We also show that the same holds if c ≡ -1 (mod a).
4
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Generators and integral points on twists of the Fermat cubic

64%
Fujita Y., Nara T.
Acta Arithmetica
|
2015
|
tom 168
|
nr 1
1-16
EN
We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.
5
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Generators and integer points on the elliptic curve y² = x³ - nx

64%
Fujita Y., Terai N.
Acta Arithmetica
|
2013
|
tom 160
|
nr 4
333-348
EN
Let E be an elliptic curve over the rationals ℚ given by y² = x³ - nx with a positive integer n. We consider first the case where n = N² for a square-free integer N. Then we show that if the Mordell-Weil group E(ℚ ) has rank one, there exist at most 17 integer points on E. Moreover, we show that for some parameterized N a certain point P can be in a system of generators for E(ℚ ), and we determine the integer points in the group generated by the point P and the torsion points. Secondly, we consider the case where n = s⁴ + t⁴ for distinct positive integers s and t. We then show that if n is fourth-power-free, the points P₁ = (-t²,s²t) and P₂ = (-s²,st²) can be in a system of generators for E(ℚ ). Furthermore, we prove that if n is square-free, then there exist at most nine integer points in the group Γ generated by the points P₁, P₂ and the torsion point (0,0). In particular, in case n = s⁴ + 1 the group Γ has exactly seven integer points.
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