# 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

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

or

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

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.

Posted on Categories Articles