If $\{a_n\}_{n=1}^\infty$ is a sequence of integers, then the definition of *primitive divisor* of one of its terms is quite natural:

**Def. 1.** A prime number $p$ is a primitive divisor of $a_n$ if $p \mid a_n$, but $p \nmid a_1 \cdots a_{n-1}$ (see [1]).

However, when Lucas sequences are involved, usually a different (unnatural, in my opinion) definition is used. Let $\{u_n\}_{n \geq 1}$ be a Lucas sequence of the first kind, with discriminant $\Delta$.

**Def. 2.** A prime number $p$ is a primitive divisor of $u_n$ if $p \mid u_n$, but $p \nmid \Delta u_1 \cdots u_{n-1}$ (see [2]).

My question is: why include $\Delta$ in Def. 2 matters?

Note that Def. 1 and Def. 2 are actually different. For example, if $\{F_n\}_{n \geq 1}$ is the (Lucas) sequence of Fibonacci numbers, then $5$ is a primitive divisor of $F_5 = 5$, respect Def. 1, but it is not respect to Def. 2, since $\Delta = 5$ for the Fibonacci's.

Thank you in advance for any elucidation.

[1] http://mathworld.wolfram.com/PrimitivePrimeFactor.html

[2] Yu Bilu , G. Hanrot , P. M. Voutier. *Existence of primitive divisors of Lucas and Lehmer numbers*. J. Reine Angew. Math (2011).