---
title: Lehmer Pairs in Number Theory
url: https://www.emergentmind.com/topics/lehmer-pairs
type: topic
---

# Lehmer Pairs in Number Theory

Lehmer pairs are a polysemous term in number theory. In the arithmetic of linear recurrences, a Lehmer pair is the algebraic input \((\alpha,\beta)\) from which a Lehmer sequence is formed; primitive-divisor theory asks when the corresponding terms acquire genuinely new prime divisors [1211.3108]. In analytic number theory, under the Riemann Hypothesis, a Lehmer pair is a pair of unusually close consecutive zeros of \(\zeta(s)\), or equivalently of Hardy’s \(Z\)-function, defined by a specific small-gap inequality and tied to the de Bruijn–Newman constant [1508.05870]. A third, terminologically adjacent but distinct usage is the Lehmer property, meaning \(\varphi(n)\mid n-1\) for a composite integer, which is not a theory of Lehmer pairs at all [1510.00638].

## 1. Recurrence-theoretic Lehmer pairs and Lehmer sequences

In the primitive-divisor literature, a Lehmer pair is a pair \((\alpha,\beta)\) satisfying
\[
(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},
\]
with these two integers relatively prime, and with
\[
\alpha/\beta \text{ not a root of unity.}
\]
This is the Lehmer analogue of a Lucas pair, where one assumes \(\alpha+\beta\in\mathbf Z\setminus\{0\}\) rather than only \((\alpha+\beta)^2\in\mathbf Z\setminus\{0\}\) [1211.3108].

The associated Lehmer sequence \((u_n)_{n\ge 0}\) is defined by the parity-dependent normalization
\[
u_n=
\begin{cases}
\dfrac{\alpha^n-\beta^n}{\alpha-\beta}, & n \text{ odd},\\[1.2ex]
\dfrac{\alpha^n-\beta^n}{\alpha^2-\beta^2}, & n \text{ even}.
\end{cases}
\]
This parity split is essential. In the Lucas case, integrality properties come directly from \(\alpha+\beta\in\mathbf Z\), whereas in the Lehmer case \(\alpha+\beta\) itself may fail to be integral, and the weaker condition \((\alpha+\beta)^2\in\mathbf Z\) is compensated by the even-term normalization [1211.3108].

The hypotheses are structural rather than cosmetic. The conditions \((\alpha+\beta)^2\in\mathbf Z\) and \(\alpha\beta\in\mathbf Z\) are what make the normalized terms behave arithmetically like integer sequences. The relative-primality condition prevents fixed common factors from being built into every term. The non-root-of-unity condition excludes the degenerate case in which \(\alpha^n-\beta^n\) vanishes for infinitely many \(n\) or becomes multiplicatively periodic [1211.3108].

Voutier’s computational work also uses integral parameters attached to a Lehmer pair. Writing
\[
x=(\alpha+\beta)^2,\qquad y=\alpha\beta,
\]
one expresses the relevant cyclotomic factors \(\Phi_n(\alpha,\beta)\) as binary forms \(F_n(x,y)\in\mathbf Z[x,y]\). This is the bridge from Lehmer pairs to explicit Thue equations [1201.6659].

## 2. Primitive divisors, defectivity, and equivalence

For a Lehmer sequence, a prime \(p\) is called a primitive divisor of the \(n\)-th Lehmer number \(u_n\) if
\[
p\mid u_n
\]
but
\[
p\nmid (\alpha^2-\beta^2)^2u_3u_4\cdots u_{n-1}.
\]
The excluded factor \((\alpha^2-\beta^2)^2\) reflects the parity-dependent denominator structure of Lehmer sequences and distinguishes the definition from the Lucas-sequence case [1211.3108].

A Lehmer pair is called \(n\)-defective when its \(n\)-th Lehmer number has no primitive divisor. This terminology is central in the small-\(n\) classification problem, where one seeks all exceptional pairs for which primitive divisors fail to appear [2411.07909].

The 2024 correction note makes the equivalence relation explicit. Two Lehmer pairs \((\alpha_1,\beta_1)\) and \((\alpha_2,\beta_2)\) are equivalent if
\[
\alpha_1/\alpha_2=\beta_1/\beta_2\in\{\pm1,\pm i\}.
\]
Equivalent pairs generate the same defectivity phenomenon up to multiplication of sequence values by a unit in \(\{\pm1,\pm i\}\). The note also stresses that the necessary condition
\[
\alpha_1\beta_1=\pm \alpha_2\beta_2
\]
is not sufficient for equivalence [2411.07909].

For classification purposes, Voutier writes Lehmer pairs in the form
\[
\left(\frac{\sqrt a-\sqrt b}{2},\,\frac{\sqrt a+\sqrt b}{2}\right),
\]
where \(a=(\alpha+\beta)^2\) and \(b=a-4\alpha\beta\). In this parametrization,
\[
(\alpha+\beta)^2=a,\qquad \alpha\beta=\frac{a-b}{4}.
\]
This encoding is used to normalize exceptional families and to compare tables across the earlier literature [2411.07909].

## 3. Small-\(n\) classification and its corrections

Stewart reduced the problem of determining all Lucas and Lehmer sequences whose \(n\)-th element does not have a primitive divisor to solving finitely many Thue equations
\[
F_n(X,Y)=m.
\]
Voutier then used the Tzanakis–de Weger method to solve the relevant equations and proved that for
\[
6<n\le 30,\qquad n\neq 8,10,12,
\]
Table 2 gives a complete list, up to multiplication of \(\alpha\) and \(\beta\) by a fourth root of unity, of all Lehmer sequences whose \(n\)-th element has no primitive divisor [1201.6659].

The small values \(n=8,10,12\) are exceptional in a different sense. For
\[
n=5,8,10,12,
\]
the associated form \(F_n(X,Y)=m\) is of total degree two and reducible to a Pell equation, so there are infinitely many solutions if any exist. In the Lucas setting the extra condition \(\alpha+\beta\in\mathbf Z\) restores finiteness, but for Lehmer sequences that restriction is absent; consequently, for \(n=8,10,12\) there are infinitely many defective Lehmer sequences [1201.6659].

The later note “\(n\)-defective Lehmer pairs for small \(n\): Corrections and Clarifications” corrects the small-\(n\) tables for
\[
n\in\{3,4,5,6,8,10,12\}.
\]
Its main corrections are concrete. For \(n=5\), it restores a missing defective pair corresponding to
\[
(p,q)=(-1,1),
\]
equivalently
\[
(\alpha,\beta)=\left(\frac{\sqrt{-1}+\sqrt{-5}}{2},\frac{\sqrt{-1}-\sqrt{-5}}{2}\right),
\]
for which
\[
\tilde u_n(\alpha,\beta)=0,1,1,-2,-3,5\qquad (n=0,1,2,3,4,5),
\]
and
\[
(\alpha^2-\beta^2)^2=5,
\]
so every prime divisor of \(\tilde u_5\) already divides the exceptional factor [2411.07909].

The same note adds the corresponding missing \(10\)-defective case, restores the missing \(12\)-defective pair
\[
(a,b)=(1,5),
\]
and removes invalid or duplicate parameter values in the \(n=3,4,5,6,10,12\) families. It also clarifies that some earlier lists conflated equivalent pairs or included cases for which \(a=(\alpha+\beta)^2=0\), so that the defining Lehmer-pair conditions failed outright [2411.07909].

## 4. Uniform primitive-divisor theorems

The modern global theorem is Voutier’s 2012 result:
\[
\text{For all } n>30{,}030,\ \text{the \(n\)-th element of any Lucas or Lehmer sequence has a primitive divisor.}
\]
For Lehmer pairs, the consequence is immediate: if \((\alpha,\beta)\) is a Lehmer pair and \((u_n)\) its associated Lehmer sequence, then every term with
\[
n>30{,}030
\]
has a primitive divisor. The bound is uniform and independent of the pair [1211.3108].

This theorem sits on top of a sequence of earlier reductions. Stewart had proved existence of an absolute bound and supplied very large explicit constants; Voutier’s earlier work had reduced the universal threshold to
\[
2\cdot 10^{10};
\]
the 2012 theorem brought it down to \(30{,}030\) [1211.3108].

A sharper result is available under a height restriction. If \((\alpha,\beta)\) generate a Lucas or Lehmer sequence with
\[
h(\beta/\alpha)\le 4,
\]
then for all
\[
n>30,
\]
the \(n\)-th element has a primitive divisor [1211.3107]. This matches the conjectural threshold proposed after the complete classification up to \(n=30\), namely that for
\[
n>30
\]
the \(n\)-th element of a Lucas or Lehmer sequence should always have a primitive divisor [1201.6659].

The proof architecture is explicit. A key factorization uses the homogeneous cyclotomic polynomial \(\Phi_n(\alpha,\beta)\), together with Stewart’s criterion that
\[
|\Phi_n(\alpha,\beta)|>n \Rightarrow u_n \text{ has a primitive divisor.}
\]
The argument then combines lower bounds for \(|\Phi_n(\alpha,\beta)|\), explicit linear forms in two logarithms, arithmetic estimates involving \(\varphi(n)\) and \(\omega(n)\), and finite computation on the remaining ranges [1211.3108].

The 2012 height-restricted paper gives an effective computational reduction. It parametrizes \(\alpha,\beta\) by integers \(p,q\), studies approximations of
\[
\frac{1}{2\pi}\arccos\!\left(\frac{p}{2q}\right)
\]
by convergents \(k/n\), and uses continued fractions to reduce a priori huge searches to finitely many candidate indices. In the difficult complex case \(|p|<2q\), this turns primitive-divisor detection into an explicit finite verification problem [1211.3107].

## 5. Lehmer pairs of zeta zeros

Under the Riemann Hypothesis, the term Lehmer pair has a different technical meaning. Let
\[
0<\gamma_-<\gamma_+
\]
be consecutive simple positive zeros of \(\Xi(t)\), let
\[
\Delta=\gamma_+-\gamma_-,
\]
and define
\[
g=\sum_{\gamma\ne\gamma_-,\gamma_+}\frac{1}{(\gamma-\gamma_-)^2}+\frac{1}{(\gamma-\gamma_+)^2}.
\]
Then \(\{\gamma_-,\gamma_+\}\) is a Lehmer pair if
\[
\Delta^2 g<\frac45.
\]
This is the Csordas–Smith–Varga condition used in the literature linking close zero pairs to the de Bruijn–Newman constant \(\Lambda\) [1508.05870].

Its significance is explicit. If \(\{\gamma_-,\gamma_+\}\) is such a pair, then one obtains a lower bound
\[
\lambda=\frac{(1-5\Delta^2 g/4)^{4/5}-1}{8g}, \qquad -\frac1{8g}<\lambda<0,
\]
with
\[
\lambda\le \Lambda.
\]
Moreover, the existence of infinitely many Lehmer pairs implies
\[
\Lambda=0.
\]
This is why close zero pairs are central in Newman-type formulations of the Riemann Hypothesis [1508.05870].

Stopple introduced the stronger notion of a strong Lehmer pair using the pre-Schwarzian derivative
\[
Pf(z)=\frac{f''(z)}{f'(z)}.
\]
A pair is strong if
\[
-\Delta^2\left(P\Xi^\prime(\gamma_+)+P\Xi^\prime(\gamma_-)\right)<\frac{42}{5},
\]
and Theorem 1 in that paper shows that strong Lehmer pairs are Lehmer pairs [1508.05870].

Simonič gave a derivative-based criterion in terms of Hardy’s \(Z\)-function. Defining
\[
\hat F(t):= -\frac13\left(\frac{Z'''}{Z'}\right)(t) +\frac14\left(\frac{Z''}{Z'}\right)^2(t)
\]
and
\[
\hat g_{\{\gamma_1,\gamma_2\}}:= \frac{(\gamma_1-\gamma_2)^2}{3}\bigl(\hat F(\gamma_1)+\hat F(\gamma_2)\bigr)-2,
\]
he proved, under the Riemann Hypothesis,
\[
0< \hat g_{\{\gamma_1,\gamma_2\}}-\bar g_{\{\gamma_1,\gamma_2\}} < 3(\gamma_1-\gamma_2)^2\left(\frac1{\gamma_1^2}+\frac1{\gamma_2^2}\right),
\]
so \(\hat g\) is a slight overestimate of the classical quantity. In particular, if
\[
\hat g_{\{\gamma_-,\gamma_+\}} < \frac45 - 3(\gamma_+-\gamma_-)^2\left(\frac1{\gamma_-^2}+\frac1{\gamma_+^2}\right),
\]
then \(\{\gamma_-,\gamma_+\}\) is a Lehmer pair [1612.08627].

Both analytic papers also supply numerical evidence. Around height \(10^6\), Stopple examined
\[
114{,}661
\]
consecutive pairs of zeros of \(\zeta(s)\), finding
\[
7398
\]
Lehmer pairs and
\[
855
\]
strong Lehmer pairs [1508.05870]. Simonič reports that among the first two million zeros there are
\[
4637
\]
pairs satisfying
\[
\gamma_{n+1}-\gamma_n<\frac{1}{\log\gamma_n},
\]
and
\[
1901
\]
pairs meeting the stronger stationary-point conditions used in his criterion for \(\Lambda=0\) [1612.08627].

## 6. Distinct but related “Lehmer” notions

The expression Lehmer pair should be distinguished from the Lehmer property. In the totient-divisibility literature, a composite integer \(n\) has the Lehmer property if
\[
\varphi(n)\mid n-1.
\]
This is the sense used in papers on Pell numbers, Lucas numbers, repunits, and related sequences. Those works study Lehmer numbers, not Lehmer pairs [1510.00638].

The distinction matters in recurrence-sequence papers. “Pell Numbers with Lehmer Property” proves that there is no composite Pell number \(P_n\) such that
\[
\varphi(P_n)\mid P_n-1
\]
[1510.00638]. “Lucas Numbers with Lehmer Property” proves the analogous nonexistence result for the Lucas sequence \((L_n)\) [1508.05709]. These results concern the totient problem and are terminologically separate from Lehmer pairs in primitive-divisor theory.

There is also a modular notion of Lehmer numbers modulo a prime \(p\). In that setting, \(a\in\{1,\dots,p-1\}\) is a Lehmer number if its inverse \(\bar a\) modulo \(p\) has opposite parity, equivalently if
\[
a+\bar a \text{ is odd.}
\]
Cohen and Trudgian do not use the phrase Lehmer pairs as a formal term, but the natural object is the inverse-pair \((a,\bar a)\), and the property is symmetric under inversion [1712.03990].

These distinctions are not merely lexical. In the recurrence-theoretic usage, a Lehmer pair is algebraic data \((\alpha,\beta)\) generating a parity-normalized sequence. In the zeta-zero usage, a Lehmer pair is a close pair of consecutive zeros satisfying a small-gap inequality. In the Lehmer-property literature, the object is instead a composite integer satisfying
\[
\varphi(n)\mid n-1.
\]
This suggests that any technical discussion of Lehmer pairs requires immediate contextualization.

Source: https://www.emergentmind.com/topics/lehmer-pairs