Lehmer Pairs in Number Theory

• Lehmer pairs are defined as algebraic pairs (α, β) generating Lehmer sequences with parity-dependent normalization and specific integrality conditions.
• They are key in primitive divisor theory, providing a framework to determine when new prime factors appear and identifying n-defective cases.
• In analytic number theory, Lehmer pairs refer to unusually close consecutive zeros of the zeta function, linking them to the de Bruijn–Newman constant and the Riemann Hypothesis.

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 (Voutier, 2012). 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 (Stopple, 2015). 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 (Faye et al., 2015).

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 $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$ (Voutier, 2012).

The associated Lehmer sequence $(u_n)_{n\ge 0}$ is defined by the parity-dependent normalization

$\zeta(s)$0

This parity split is essential. In the Lucas case, integrality properties come directly from $\zeta(s)$1, whereas in the Lehmer case $\zeta(s)$2 itself may fail to be integral, and the weaker condition $\zeta(s)$3 is compensated by the even-term normalization (Voutier, 2012).

The hypotheses are structural rather than cosmetic. The conditions $\zeta(s)$4 and $\zeta(s)$5 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 $\zeta(s)$6 vanishes for infinitely many $\zeta(s)$7 or becomes multiplicatively periodic (Voutier, 2012).

Voutier’s computational work also uses integral parameters attached to a Lehmer pair. Writing

$\zeta(s)$8

one expresses the relevant cyclotomic factors $\zeta(s)$9 as binary forms $Z$0. This is the bridge from Lehmer pairs to explicit Thue equations (Voutier, 2012).

2. Primitive divisors, defectivity, and equivalence

For a Lehmer sequence, a prime $Z$1 is called a primitive divisor of the $Z$2-th Lehmer number $Z$3 if

$Z$4

but

$Z$5

The excluded factor $Z$6 reflects the parity-dependent denominator structure of Lehmer sequences and distinguishes the definition from the Lucas-sequence case (Voutier, 2012).

A Lehmer pair is called $Z$7-defective when its $Z$8-th Lehmer number has no primitive divisor. This terminology is central in the small-$Z$9 classification problem, where one seeks all exceptional pairs for which primitive divisors fail to appear (Voutier, 2024).

The 2024 correction note makes the equivalence relation explicit. Two Lehmer pairs $\varphi(n)\mid n-1$0 and $\varphi(n)\mid n-1$1 are equivalent if

$\varphi(n)\mid n-1$2

Equivalent pairs generate the same defectivity phenomenon up to multiplication of sequence values by a unit in $\varphi(n)\mid n-1$3. The note also stresses that the necessary condition

$\varphi(n)\mid n-1$4

is not sufficient for equivalence (Voutier, 2024).

For classification purposes, Voutier writes Lehmer pairs in the form

$\varphi(n)\mid n-1$5

where $\varphi(n)\mid n-1$6 and $\varphi(n)\mid n-1$7. In this parametrization,

$\varphi(n)\mid n-1$8

This encoding is used to normalize exceptional families and to compare tables across the earlier literature (Voutier, 2024).

3. Small-$\varphi(n)\mid n-1$9 classification and its corrections

Stewart reduced the problem of determining all Lucas and Lehmer sequences whose $(\alpha,\beta)$0-th element does not have a primitive divisor to solving finitely many Thue equations

$(\alpha,\beta)$1

Voutier then used the Tzanakis–de Weger method to solve the relevant equations and proved that for

$(\alpha,\beta)$2

Table 2 gives a complete list, up to multiplication of $(\alpha,\beta)$3 and $(\alpha,\beta)$4 by a fourth root of unity, of all Lehmer sequences whose $(\alpha,\beta)$5-th element has no primitive divisor (Voutier, 2012).

The small values $(\alpha,\beta)$6 are exceptional in a different sense. For

$(\alpha,\beta)$7

the associated form $(\alpha,\beta)$8 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)$9 restores finiteness, but for Lehmer sequences that restriction is absent; consequently, for $$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$0 there are infinitely many defective Lehmer sequences (Voutier, 2012).

The later note “$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$1-defective Lehmer pairs for small $$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$2: Corrections and Clarifications” corrects the small-$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$3 tables for

$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$4

Its main corrections are concrete. For $$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$5, it restores a missing defective pair corresponding to

$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$6

equivalently

$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$7

for which

$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$8

and

$$(\alpha+\beta)^2\in \mathbf Z\setminus\{0\},\qquad \alpha\beta\in \mathbf Z\setminus\{0\},$$9

so every prime divisor of $\alpha/\beta \text{ not a root of unity.}$0 already divides the exceptional factor (Voutier, 2024).

The same note adds the corresponding missing $\alpha/\beta \text{ not a root of unity.}$1-defective case, restores the missing $\alpha/\beta \text{ not a root of unity.}$2-defective pair

$\alpha/\beta \text{ not a root of unity.}$3

and removes invalid or duplicate parameter values in the $\alpha/\beta \text{ not a root of unity.}$4 families. It also clarifies that some earlier lists conflated equivalent pairs or included cases for which $\alpha/\beta \text{ not a root of unity.}$5, so that the defining Lehmer-pair conditions failed outright (Voutier, 2024).

4. Uniform primitive-divisor theorems

The modern global theorem is Voutier’s 2012 result: $\alpha/\beta \text{ not a root of unity.}$6 For Lehmer pairs, the consequence is immediate: if $\alpha/\beta \text{ not a root of unity.}$7 is a Lehmer pair and $\alpha/\beta \text{ not a root of unity.}$8 its associated Lehmer sequence, then every term with

$\alpha/\beta \text{ not a root of unity.}$9

has a primitive divisor. The bound is uniform and independent of the pair (Voutier, 2012).

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

$\alpha+\beta\in\mathbf Z\setminus\{0\}$0

the 2012 theorem brought it down to $\alpha+\beta\in\mathbf Z\setminus\{0\}$1 (Voutier, 2012).

A sharper result is available under a height restriction. If $\alpha+\beta\in\mathbf Z\setminus\{0\}$2 generate a Lucas or Lehmer sequence with

$\alpha+\beta\in\mathbf Z\setminus\{0\}$3

then for all

$\alpha+\beta\in\mathbf Z\setminus\{0\}$4

the $\alpha+\beta\in\mathbf Z\setminus\{0\}$5-th element has a primitive divisor (Voutier, 2012). This matches the conjectural threshold proposed after the complete classification up to $\alpha+\beta\in\mathbf Z\setminus\{0\}$6, namely that for

$\alpha+\beta\in\mathbf Z\setminus\{0\}$7

the $\alpha+\beta\in\mathbf Z\setminus\{0\}$8-th element of a Lucas or Lehmer sequence should always have a primitive divisor (Voutier, 2012).

The proof architecture is explicit. A key factorization uses the homogeneous cyclotomic polynomial $\alpha+\beta\in\mathbf Z\setminus\{0\}$9, together with Stewart’s criterion that

$(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$0

The argument then combines lower bounds for $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$1, explicit linear forms in two logarithms, arithmetic estimates involving $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$2 and $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$3, and finite computation on the remaining ranges (Voutier, 2012).

The 2012 height-restricted paper gives an effective computational reduction. It parametrizes $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$4 by integers $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$5, studies approximations of

$(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$6

by convergents $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$7, and uses continued fractions to reduce a priori huge searches to finitely many candidate indices. In the difficult complex case $(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$8, this turns primitive-divisor detection into an explicit finite verification problem (Voutier, 2012).

5. Lehmer pairs of zeta zeros

Under the Riemann Hypothesis, the term Lehmer pair has a different technical meaning. Let

$(\alpha+\beta)^2\in\mathbf Z\setminus\{0\}$9

be consecutive simple positive zeros of $(u_n)_{n\ge 0}$0, let

$(u_n)_{n\ge 0}$1

and define

$(u_n)_{n\ge 0}$2

Then $(u_n)_{n\ge 0}$3 is a Lehmer pair if

$(u_n)_{n\ge 0}$4

This is the Csordas–Smith–Varga condition used in the literature linking close zero pairs to the de Bruijn–Newman constant $(u_n)_{n\ge 0}$5 (Stopple, 2015).

Its significance is explicit. If $(u_n)_{n\ge 0}$6 is such a pair, then one obtains a lower bound

$(u_n)_{n\ge 0}$7

with

$(u_n)_{n\ge 0}$8

Moreover, the existence of infinitely many Lehmer pairs implies

$(u_n)_{n\ge 0}$9

This is why close zero pairs are central in Newman-type formulations of the Riemann Hypothesis (Stopple, 2015).

Stopple introduced the stronger notion of a strong Lehmer pair using the pre-Schwarzian derivative

$\zeta(s)$00

A pair is strong if

$\zeta(s)$01

and Theorem 1 in that paper shows that strong Lehmer pairs are Lehmer pairs (Stopple, 2015).

Simonič gave a derivative-based criterion in terms of Hardy’s $\zeta(s)$02-function. Defining

$\zeta(s)$03

and

$\zeta(s)$04

he proved, under the Riemann Hypothesis,

$\zeta(s)$05

so $\zeta(s)$06 is a slight overestimate of the classical quantity. In particular, if

$\zeta(s)$07

then $\zeta(s)$08 is a Lehmer pair (Simonič, 2016).

Both analytic papers also supply numerical evidence. Around height $\zeta(s)$09, Stopple examined

$\zeta(s)$10

consecutive pairs of zeros of $\zeta(s)$11, finding

$\zeta(s)$12

Lehmer pairs and

$\zeta(s)$13

strong Lehmer pairs (Stopple, 2015). Simonič reports that among the first two million zeros there are

$\zeta(s)$14

pairs satisfying

$\zeta(s)$15

and

$\zeta(s)$16

pairs meeting the stronger stationary-point conditions used in his criterion for $\zeta(s)$17 (Simonič, 2016).

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 $\zeta(s)$18 has the Lehmer property if

$\zeta(s)$19

This is the sense used in papers on Pell numbers, Lucas numbers, repunits, and related sequences. Those works study Lehmer numbers, not Lehmer pairs (Faye et al., 2015).

The distinction matters in recurrence-sequence papers. “Pell Numbers with Lehmer Property” proves that there is no composite Pell number $\zeta(s)$20 such that

$\zeta(s)$21

(Faye et al., 2015). “Lucas Numbers with Lehmer Property” proves the analogous nonexistence result for the Lucas sequence $\zeta(s)$22 (Faye et al., 2015). 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 $\zeta(s)$23. In that setting, $\zeta(s)$24 is a Lehmer number if its inverse $\zeta(s)$25 modulo $\zeta(s)$26 has opposite parity, equivalently if

$\zeta(s)$27

Cohen and Trudgian do not use the phrase Lehmer pairs as a formal term, but the natural object is the inverse-pair $\zeta(s)$28, and the property is symmetric under inversion (Cohen et al., 2017).

These distinctions are not merely lexical. In the recurrence-theoretic usage, a Lehmer pair is algebraic data $\zeta(s)$29 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

$\zeta(s)$30

This suggests that any technical discussion of Lehmer pairs requires immediate contextualization.