The extension of the $D(-k)$-pair $\{k,k+1\}$ to a quadruple
Let $k$ be a positive integer. In this paper, we prove that if $\{k,k+1,c,d\}$ is a $D(-k)$-quadruple with $c>1$, then $d=1$.
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arXiv subjects
Yasutsugu Fujita
Publications and source records attributed to Yasutsugu Fujita.
The Mordell-Weil bases for the elliptic curve of the form $\boldsymbol{y^2=x^3-m^2x+n^2}$
Let $E_{m,n}$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^2x+n^2$, where $m$ and $n$ are positive integers. Brown and Myers showed that the curve $E_{1,n}$ has rank at least two for all $n$. In the present paper, we specify the two points which can be extended to a basis for $E_{1,n}(\mathbb{Q})$ under certain conditions described explicitly. Moreover, we verify a similar result for the curve $E_{m,1}$, which, however, gives a basis for the rank three part of $E_{m,1}(\mathbb{Q})$.
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Two-parameter families of uniquely extendable Diophantine triples
Let A,K be positive integers and u=-2,-1,1 or 2. The main contribution of the paper is a proof that each of the D(u^2)-triples {K,A^2K+2uA,(A+1)^2K+2u(A+1)} has unique extension to a D(u^2)-quadruple.
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On the Mordell--Weil group of the elliptic curve y^2=x^3+n
We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mordell--Weil group.
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