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Primitive divisors of certain elliptic divisibility sequences. (English) Zbl 1241.11037

For a non-torsion rational point \(P=(x(P),y(nP))\) on an elliptic curve \(E:y^2=x^3+ax+b\) there is an associated elliptic divisibility sequence \((B_n(P))\) defined by \(x(nP)=A_n(P)/B_n(P)^2\) in lowest terms. J. H. Silverman [J. Number Theory 30, No. 2, 226–237 (1988; Zbl 0654.10019)] showed that any such sequence has a Zsigmondy bound – that is, there is some \(N=N(P,E)\) with the property that for any \(n\geq N\) the term \(B_n\) has a primitive divisor (a prime divisor of \(B_n\) that is not a divisor of any earlier term). For specific curves, work of G. Everest, G. McLaren and the reviewer [J. Number Theory 118, No. 1, 71–89 (2006; Zbl 1093.11038)] gave a uniform small bound for congruent number curves, and P. Ingram [J. Number Theory 123, No. 2, 473–486 (2007; Zbl 1170.11010)] extended and improved these results. A key role is played by the doubling map on the curve, and here the question of Zsigmondy bounds for the sequence obtained by repeated doubling (rather than repeated addition of the point) is studied. For the curve \(E:y^2=x^3+D\), where \(D\) is a sixth-power-free integer, is is shown that every term in the sequence \((B_{2^m}(P))\) with \(m\geq3\) has a primitive divisor. This is applied to give a new method for solving special cases of the associated Diophantine equation \(y^2=x^3+d^n\).

MSC:

11D61 Exponential Diophantine equations
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
11D45 Counting solutions of Diophantine equations
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