156 research outputs found
On Mordell-Weil groups of elliptic curves induced by Diophantine triples
We study the possible structure of the groups of rational points on elliptic
curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals
such that the product of any two of them is one less than a square.Comment: 17 pages; to appear in Glasnik Matematicki 42 (2007
Bounds for the size of sets with the property D(n)
Let n be a nonzero integer and a_1 < a_2 < ... <a_m positive integers such
that a_i*a_j + n is a perfect square for all 1 <= i < j <= m. It is known that
m <= 5 for n = 1. In this paper we prove that m <= 31 for |n| <= 400 and m <
15.476 log|n| for |n| > 400.Comment: 9 pages. Revised - removed a gap in the proof of Proposition 1; to
appear in Glasnik Matematick
Root separation for irreducible integer polynomials
We establish new results on root separation of integer, irreducible
polynomials of degree at least four. These improve earlier bounds of Bugeaud
and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd
degree).Comment: 8 pages; revised version; to appear in Bull. Lond. Math. So
Rank zero elliptic curves induced by rational Diophantine triples
Rational Diophantine triples, i.e. rationals a,b,c with the property that
ab+1, ac+1, bc+1 are perfect squares, are often used in construction of
elliptic curves with high rank. In this paper, we consider the opposite problem
and ask how small can be the rank of elliptic curves induced by rational
Diophantine triples. It is easy to find rational Diophantine triples with
elements with mixed signs which induce elliptic curves with rank 0. However,
the problem of finding such examples of rational Diophantine triples with
positive elements is much more challenging, and we will provide the first such
known example.Comment: 7 page
Root separation for reducible monic polynomials of odd degree
We study root separation of reducible monic integer polynomials of odd
degree. Let h(P) be the naive height, sep(P) the minimal distance between two
distinct roots of an integer polynomial P(x) and sep(P)=h(P)^{-e(P)}. Let
e_r*(d)=limsup_{deg(P)=d, h(P)-> +infty} e(P), where the limsup is taken over
the reducible monic integer polynomials P(x) of degree d. We prove that e_r*(d)
<= d-2. We also obtain a lower bound for e_r*(d) for d odd, which improves
previously known lower bounds for e_r*(d) when d = 5, 7, 9.Comment: 8 pages, to appear in Rad Hrvat. Akad. Znan. Umjet. Mat. Zna
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