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](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehmer%20numbers) 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](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehmer%20numbers))


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](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehmer%20numbers).


A *Lehmer pair* is a pair of complex numbers $(\alpha, \beta)$ such that $(\alpha + \beta)^2$ and $\alpha\beta$ 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](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehmer%20numbers) 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


$$(1) \qquad (a, b) = (\phi\_{k-2\varepsilon}, \phi\_{k-2\varepsilon} - 4\phi\_k) \quad (k \geq 3)$$


or


$$(2) \qquad (a, b) = (\psi\_{k-2\varepsilon}, \psi\_{k-2\varepsilon} - 4\psi\_k) \quad (k \neq 1) ,$$


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 $\alpha\_0\beta\_0 = -1$ 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 $(a - b) / 4 = \alpha\beta = \pm \alpha\_0 \beta\_0 = \pm 1$ so that $a - b = \pm 4$. In case (1), we have $a - b = 4\phi\_k \geq 8$, because $k \geq 3$. In case (2), we have $a - b = 4\phi\_k \geq 8$, 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](https://www.google.com/search?q=Bilu%2C%20Hanrot%2C%20and%20Voutier%2C%20Existence%20of%20primitive%20divisors%20of%20Lucas%20and%20Lehmer%20numbers). 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.



Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)
Yes, there are some omissions in the lists in the original BHV article. I think all of them were fixed by Mourad Abouzaid


Mourad Abouzaid, Les nombres de Lucas et Lehmer sans diviseur primitif, J. Théor. Nombres Bordeaux 18 (2006), no. 2, 299–313.


