nt.number theory – Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)

I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in:

1 Bilu, Hanrot, and Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, Journal für die reine und angewandte Mathematik (Crelles Journal), 2001. (available on the web)

Since I might very much be wrong myself, I would like to know if my reasoning is correct or not. I recall below the main definitions of 1.

A Lehmer pair is a pair of complex numbers $(alpha, beta)$ such that $(alpha + beta)^2$ and $alphabeta$ are non-zero coprime integers and $alpha / beta$ is not a root of unity. Two Lehmer pairs $(alpha_1, beta_1)$ and $(alpha_2, beta_2)$ are said to be equivalent if $alpha_1 / alpha_2 = beta_1 / beta_2 in {-1,+1,sqrt{-1},-sqrt{-1}}$. Given a Lehmer pair $(alpha, beta)$, the associated Lehmer sequence is
$$widetilde{u}_n(alpha, beta) := begin{cases} (alpha^n – beta^n) / (alpha – beta) & text{ if $n$ is odd} \
(alpha^n – beta^n) / (alpha^2 – beta^2) & text{ if $n$ is even}end{cases}$$

for every positive integer $n$ (it is an integer sequence).

A prime number $p$ is a primitive divisor of $widetilde{u}_n(alpha, beta)$ if $p$ divides $widetilde{u}_n(alpha, beta)$ but does not divide $(alpha^2 – beta^2)^2 widetilde{u}_1(alpha, beta) cdots widetilde{u}_{n-1}(alpha, beta)$. If $widetilde{u}_n(alpha, beta)$ has no primitive divisor then the Lehmer pair $(alpha, beta)$ is $n$-defective.

One of the claims of Theorem 1.3 of 1 is that, up to equivalence, all $5$-defective Lehmer pairs are of the form $((sqrt{a} – sqrt{b})/2, (sqrt{a} + sqrt{b})/2)$ with

$$(1) qquad (a, b) = (phi_{k-2varepsilon}, phi_{k-2varepsilon} – 4phi_k) quad (k geq 3)$$


$$(2) qquad (a, b) = (psi_{k-2varepsilon}, psi_{k-2varepsilon} – 4psi_k) quad (k neq 1) ,$$

where $k$ is a nonnegative integer, $varepsilon in {-1, +1}$, $(phi_n)$ is the sequence of Fibonacci numbers, and $(psi_n)$ is the sequence of Lucas numbers.

Claim 1: The Lehmer pair $(alpha_0, beta_0) := ((1 – sqrt{5}) / 2, (1 + sqrt{5}) / 2)$ is $5$-defective.

First, note that $(alpha_0 + beta_0)^2 = 1$ and $alpha_0beta_0 = -1$ are non-zero coprime integers and $alpha_0 / beta_0 = (sqrt{5}-3) / 2$ is not a root of unity, so that $(alpha_0, beta_0)$ is indeed a Lehmer pair. Second, for the associated Lehmer sequence we have $widetilde{u}_5 = 5$ and $(alpha_0^2 – beta_0^2)^2 = 5$, thus $widetilde{u}_5$ has no primitive divisor and $(alpha_0, beta_0)$ is $5$-defective.

Claim 2: The Lehmer pair $(alpha_0, beta_0)$ is not equivalent to a pair of the form $(alpha, beta) = ((sqrt{a} – sqrt{b})/2, (sqrt{a} + sqrt{b})/2)$ with $(a, b)$ as in (1) or (2).

For the sake of contradiction suppose $(alpha_0, beta_0)$ is equivalent to a pair of the form $(alpha, beta) = ((sqrt{a} – sqrt{b})/2, (sqrt{a} + sqrt{b})/2)$ with $(a, b)$ as in (1) or (2). Then $(a – b) / 4 = alphabeta = pm alpha_0 beta_0 = pm 1$ so that $a – b = pm 4$. In case (1), we have $a – b = 4phi_k geq 8$, because $k geq 3$. In case (2), we have $a – b = 4phi_k geq 8$, because $k neq 1$. Absurd.

Possible source of the error: I think that the missing $5$-defective pair is lost in the last paragraph of case $n = 5$ in section “Small $n$” of 1. It is said that:

“By (28), we have $k geq 3$ in the case (34), and $k neq 1$ in the case (35).”

But, in case (35), $k = 1$ (and $varepsilon = 1$) are not in contradiction with (28).

In other words, (2) should allow $k = 1$ (and $varepsilon = 1$).
This would lead to the $5$-defective pair $((sqrt{-1} + sqrt{-5}) / 2, (sqrt{-1} – sqrt{-5}) / 2)$, which is equivalent to $(alpha_0, beta_0)$.

Thank in advance to anyone who takes the time to check.