Solving elliptic diophantine equations avoiding Thue equations and elliptic logarithms.
de Weger, Benjamin M.M. (1998)
Experimental Mathematics
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de Weger, Benjamin M.M. (1998)
Experimental Mathematics
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Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)
Colloquium Mathematicae
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We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions. ...
Stroeker, Roel J., de Weger, Benjamin M.M. (1994)
Experimental Mathematics
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Stroeker, Roel J., Tzanakis, Nikos (1999)
Experimental Mathematics
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Maciej Ulas (2007)
Colloquium Mathematicae
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Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
Wai Yan Pong, Roelof J. Stroeker (2012)
Acta Arithmetica
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S. Akhtari, A. Togbé, P. G. Walsh (2008)
Acta Arithmetica
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Levesque, C. (2003)
International Journal of Mathematics and Mathematical Sciences
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H. Kleiman (1976)
Journal für die reine und angewandte Mathematik
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Yang Hai, P. G. Walsh (2010)
Acta Arithmetica
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Ernst, Bruno (1996)
General Mathematics
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Alan Filipin (2009)
Acta Arithmetica
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Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)
Journal de Théorie des Nombres de Bordeaux
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We prove that Hilbert’s Tenth Problem for a ring of integers in a number field has a negative answer if satisfies two arithmetical conditions (existence of a so-called set of integers and of an elliptic curve of rank one over ). We relate division-ample sets to arithmetic of abelian varieties.
Shin-ichi Katayama, Claude Levesque (2003)
Acta Arithmetica
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W. J. Ellison (1970-1971)
Séminaire de théorie des nombres de Bordeaux
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Jianhua Chen (2001)
Acta Arithmetica
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