منابع مشابه
Elliptic binomial diophantine equations
The complete sets of solutions of the equation (n k ) = (m ` ) are determined for the cases (k, `) = (2, 3), (2, 4), (2, 6), (2, 8), (3, 4), (3, 6), (4, 6), (4, 8). In each of these cases the equation is reduced to an elliptic equation, which is solved by using linear forms in elliptic logarithms. In all but one case this is more or less routine, but in the remaining case ((k, `) = (3, 6)) we h...
متن کاملSolving Elliptic Diophantine Equations Avoiding Thue Equations and Elliptic Logarithms
This research was supported by the Netherlands Mathematical Research Foundation SWON with nancial aid from the Netherlands Organization for Scienti c Research NWO. We determine the solutions in integers of the equation y = (x + p)(x + p) for p = 167, 223, 337, 1201. The method used was suggested to us by Yu. Bilu, and is shown to be in some cases more efficient than other general purpose method...
متن کاملA Binomial Diophantine Equation
following result. THEOREM 1. The only (n,m)eZ with n^2 and m5=4 satisfying © = ( 7 ) a r e {n> m)=(2> 4)> (6> 6)> and (21> Our binomial diophantine equation represents an elliptic curve, since it can be rewritten as a quartic polynomial being a square. Indeed, on putting u = 2/i 1 and v = 2m 3, we see at once that Theorem 1 follows from the following result. THEOREM 2. The only (u, v) e Z with ...
متن کاملSome Diophantine Equations Related to Positive-rank Elliptic Curves
We give conditions on the rational numbers a, b, c which imply that there are infinitely many triples (x, y, z) of rational numbers such that x+ y + z = a+ b+ c and xyz = abc. We do the same for the equations x + y + z = a + b + c and x + y + z = a + b + c. These results rely on exhibiting families of positive-rank elliptic curves.
متن کاملDiophantine approximation and Diophantine equations
The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 1999
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-99-01047-9