## 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

### Keywords:

elliptic curve; primitive divisor; Zsigmondy’s theorem; prime
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### References:

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