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Primitive divisors of elliptic divisibility sequences. (English) Zbl 1093.11038

Let \(A=(a_n)_{n\geq 1}\) be an integer sequence. A prime number \(p\) which divides a term \(a_n\) is called a {primitive divisor} of \(a_n\) if \(p\) does not divide any term \(a_k\) with \(1\leq k<n\). One defines the Zsigmondy bound \(Z(A)\) for \(A\) by the equality \(Z(A)= \max \{ n\mid a_n\) does not have a primitive divisor\(\},\) if this set is finite, and \(Z(A)=\infty\) if not. Zsigmondy has proved in 1892 that for the Mersenne sequence \(M=(2^n-1)_{n\geq 1}\), one has \(Z(M)=6\). He also showed that for any coprime integers \(a\) and \(b\), we have \(Z((a^n-b^n)_{n\geq 1})\leq 6\). A few years ago, Bilu, Hanrot and Voutier have obtained the bound \(Z(L)\leq 30\) for any non-trivial Lucas or Lehmer sequence \(L\). In this paper the authors study to what extent this kind of results might hold for elliptic divisibility sequences.
Let us recall the definition of the elliptic divisibility sequences. Let \(E\) be an elliptic curve defined over \(\mathbb Q\) given by a minimal Weierstrass equation and \(P=(x(P),y(P))\) be a non-torsion point in \(E(\mathbb Q)\). For any integer \(n\geq 1\), let us write \[ x(nP)={a_n\over b_n}, \] in lowest terms, with \(a_n\in \mathbb Z\) and \(b_n\in \mathbb N\). The sequence \(B_{E,P}=(b_n)_{n\geq 1}\) is a divisibility sequence, which means that \(b_m\) divides \(b_n\) whenever \(m\) divides \(n\). Following a suggestion of Silverman, such sequences are called elliptic divisibility sequences. Silverman has shown in 1988 that \(Z(B_{E,P})\) is finite. The authors determine uniform explicit bounds for \(Z(B_{E,P})\), for certain infinite families of elliptic curves \(E/\mathbb Q\). Their results appear as the first examples given explicitly of the elliptic Zsigmondy theorem. Given an integer sequence \(A\), they are led to consider the even and the odd Zsigmondy bounds for \(A\). These bounds \(Z_e(A)\) and \(Z_o(A)\) are defined as \(Z(A)\), by considering only the even and the odd indexes respectively. We obviously have the equality \(Z(A)=\max(Z_e(A),Z_o(A))\). Let us mention one of the results obtained by the authors. Let \(a\) be a square-free positive integer and \(E\) be the elliptic curve defined by the equation \[ y^2=x^3-a^2x. \] Suppose that \(E\) has a non-torsion point \(P\in E(\mathbb Q)\). Then, one has \(Z_e(B_{E,P})\leq 10\). Furthermore, if \(x(P)<0\), then \(Z_o(B_{E,P})\leq 3\) and if \(x(P)\) is a square, then \(Z_o(B_{E,P})\leq 21\). In case \(a=5\), with \(P=(-4,6)\), they prove that \(Z(B_{E,P})=1\).

MSC:

11G05 Elliptic curves over global fields
11A41 Primes
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References:

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