Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

H

ANS

R

OSKAM

Prime divisors of linear recurrences and Artin’s

primitive root conjecture for number fields

Journal de Théorie des Nombres de Bordeaux, tome

13, n

o

_{1 (2001),}

p. 303-314

<http://www.numdam.org/item?id=JTNB_2001__13_1_303_0>

© Université Bordeaux 1, 2001, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Prime divisors of

linear

recurrences

and Artin’s

primitive

root

_conjecture

for number

fields

par HANS ROSKAM

RÉSUMÉ. Soit S une suite définie _par une récurrence linéaire entière d’ordre k &#x3E; 3. On note l’ensemble des nombres

pre-miers _qui divisent au moins l’un des termes de S. Nous donnons

une

_{approche heuristique}

du

_problème

selon

_lequel

PS admet ou non une densité

_naturelle,

_{et montrons que} certains aspects de

ces

_heuristiques

sont corrects. Sous

_{l’hypothèse}

d’une certaine

généralisation

de la

_conjecture

d’Artin _pour les racines

primi-tives, nous _{montrons que}

_{PS possède}

une densité _asymptotique inférieure _{pour toute} suite S

_{"générique".}

Nous donnons en illus-tration des

_{exemples numériques.}

ABSTRACT. Let S be a linear _{integer recurrent sequence} of order k ~ _3, and define _PS as the set of _primes that divide at least one term of S. We

_give

a heuristic

_approach

to the

_problem

whether _PS has a natural

_density,

and prove that part of our

heuristics is correct. Under the _assumption of a

generalization

of Artin’s _primitive root _conjecture, we find that PS has _positive lower

_density

for

_{’generic’}

_sequences S. Some numerical

_examples

are included.

1. Introduction

An

_integer

_sequence ,S‘ _

Ix,,1001

is said _to

satisfy

_a linear _recurrence of

order

_1~,

if there exist

_integers

_{al, a2, ... , a~} such that

If such a _sequence ,S’ does not

_satisfy

a linear recurrence of order smaller

than

_1~,

we _say that ~S’ is a linear _{recurrent sequence} of order l~. In this case the xn can be

_given

as an

_exponential

_expression

in the roots of the characteristic

_{polynomial f}

=

X ~ -

_E

Z[X]

of the _recurrence.

For

_example,

if

_f

is

_separable

with roots we have

where E are determined

_by

the initial values

x

n n=l

Associated to a linear

_integer

recurrent sequence ,S’ is the set

_Ps

of

_primes

dividing

the _sequence, where a

_prime

_p is said to divide the _sequence if _p divides some term of the _sequence. We _say that ,S’ is

_degenerate

if the

as-sociated characteristic

_polynomial

is either

_inseparable

or has two different

roots whose

_quotient

is a root of

_unity.

If ,S’ is

_degenerate,

the set

_Ps

is

possibly

finite. The fact that

_Ps

is infinite for

_{non-degenerate}

linear

re-current _sequences ,S’ of order 2 is often attributed to Ward

_[15].

_However,

already

in 1921

_P61ya

_[10]

_proved

that for all

_{non-degenerate}

linear

recur-rent _sequences ~S’ of order k &#x3E;

_2,

the set

_Ps

is infinite. His

_proof

shows that

a finite set

_Ps

contradicts the

_growth

rate of S.

P61ya’s

method seems

inadequate

to determine the ’size’ of

Ps.

In

par-ticular,

it does not allow us to decide whether

Ps

has a

density

inside

the set of all

_primes.

Almost all results on the existence and value of

6(PS)

are for second order linear recurrent _sequences ,S’. In this second order _case, the method of

_proof

_depends

on two

_properties

of the _sequence: whether the _{sequence is} ’torsion’

_[1,

_{4, 6,}

_9]

or non-torsion

[13],

and whether the

characteristic

_polynomial

is reducible over

_Q

[1,

_{4, 9,}

13]

or irreducible

[6].

Essentially nothing

is known for _sequences of order

_larger

than 2. The methods for second order _sequences can be made to work in some _very

special

cases: there exist

_higher

order linear _{recurrent sequences}

_having

a set of maximal

_prime

divisors of

_positive

_density

_[1].

Ward

_[16]

introduced the term maximal

_prime

divisors of a k-th order linear recurrent _sequence for those

_primes

that divide I~ - 1 consecutive terms of the _sequence, but do not divide all terms.

In this _paper we focus on linear recurrent _sequences ,S’ of order k &#x3E; 3

whose associated characteristic

_polynomial

is

_{separable. Unfortunately,}

we are not able to prove that the set

Ps

of

_primes

_dividing

such a _sequence

has a

_density.

_However,

for a

_large

class of linear recurrent _sequences we can _prove that

Ps

has

_positive

lower

_density,

if we assume a

generalization

of Artin’s

_conjecture

on

primitive

roots.

The _{paper is}

_organized

as follows. In section 2 we

give

a different char-acterization of the set

_{Ps .}

This characterization is used in section

_3,

to

give

a heuristic

approach

to the

_problem

whether the set

_Ps

contains a set of

_positive

_density.

Section 3 also contains a theorem

_stating

that

_part

of

the heuristics is correct. This

_theorem,

combined with a

_{generalization}

of

density

for

_{’generic’}

sequences S. After

proving

the theorem in section

4,

we

_give

some numerical

examples

in section 5.

2. Reformulation of the

_problem

Let S = be an

_integer

_sequence

_satisfying

the linear recurrence

of order &#x3E; 2 with

_separable

characteristic

_polynomial

The _sequence ,S’ is an element of

_{V f,}

the set of all _sequences of

algebraic

numbers

_satisfying

the linear recurrence with characteristic

polynomial f.

By

defining

addition and scalar

_{multiplication}

of _sequences

component-wise,

the set

_{V f}

becomes a

_{Q-vectorspace.}

As

f

is

separable,

the _sequences

form a

_Q-basis

for

_{V f ,}

and there exist _{cl , ... , Ck} E

_Q

such that xn -

Ciai

for all

positive integers

n. The absolute Galois

group

GQ

of

Q

permutes

the

_a2’s

and acts

_trivially

on the

_integers

_xn.

Using

that the form a basis for

_{V f,}

we find that

the set is

_{GQ-invariant.}

_Moreover,

each a E

_GQ

induces the same

permutation

on the

Now let a denote the residue class of X in the free

Q-algebra

A

-Ca ~X~~ / ( f )

of rank k.

_By

the Chinese remainder

_theorem,

the

_c2’s

determine

a

unique

c E A such that

c(ai) =

c2 for i =

1, ... ,

k. Because of the last

remark of the

_previous

_paragraph,

_GQ

acts

_trivially

on c. We conclude that

there exists a

_unique

c E

_{(~ ~X ~ / ( f )}

such that

Denote the set of

_primes

that divide the _sequence S

_by

_Ps.

A

_prime

_p is

an element of

Ps if

and

only

if there exists an

integer n

such that

In order to

_interchange

the

_trace-map

and the reduction modulo

_pZ,

we

choose an

_integer

d ~

0 such that cd E

_{Z (X ~ / ( f )}

and define R =

Z[~].

The

_(maximal)

_primes

of the

_ring

R are of the form

pR,

with _p a rational

prime

not

_dividing

d. In the

_following

we

_disregard

the

finitely

_many

_primes

dividing

d. This restriction has no effect on the existence or value of the

density

of

_Ps.

The elements can are contained in C~ =

R~X~/( f ),

a free

_R-algebra

of

rank k. As the trace is stable under base

_change

and the

_Q-algebras

A and 0 0p

Q

are

isomorphic,

we can write

equation

(2)

as

To _{compute xn} modulo a

prime p

of

R,

we tensor the _trace-map Tr :

We find that a

_prime

_p of R divides the _sequence if and

only

if there exists an

integer n

such that

where the bar means reduction modulo In other

words,

if H denotes

the kernel of the trace _map Tr :

_{Fp ,}

we have the

following

criterion

for a

_prime

_p of R to divide the _sequence:

This characterization hints to

_why

the determination of

_PS

is difficult. The

group H is an additive

_subgroup

of the

_ring

_whereas,

for almost

all

_primes,

_c(a)

is a coset of the

_{multiplicative subgroup}

_(a)

C

(O/pO)* .

Such a mixture of an additive and

multiplicative

structure is

notoriously

difficult to

_study.

3. Heuristic

_approach

In this section we let _{p range} over the

primes

of R and

predict

the

’probability’ that p

divides a fixed linear recurrent _sequence S. If this

’probability’

is

_positive,

we

expect Ps

to be a set of

_primes

of

_positive

density

inside the set of all

_primes.

_By

the

_equivalence

₍₄₎

of section

_2,

the likelihood that _p divides S will

_depend

on the size of the

subgroup

(a)

C

(O/pO)* .

In the extreme case that a

_generates

_{(O/pO)* ,}

the intersection

c(a)

nH is

_non-empty

and _p divides the _sequence. To make the

_importance

of the size of

_(a)

more

_explicit,

we fix a

_prime

_p of R such that ca is a unit

modulo

_pO.

As the

trace-map

is

_surjective,

its kernel H has index p. The

probability

for a

randomly

chosen x

EO/pO

to have non-zero trace

is

1-!.

The elements in the coset

_e(1i)

c are not

_randomly

chosen

p

elements of

_{0/ pO.}

However,

if we let _{p range} over the

primes

of

R,

it seems

plausible

that the additive and

_{multiplicative}

structure of the

_ring

are

’independent’.

Therefore,

we assume that as _p

_varies,

the

_{’probability’}

that an element in the coset

_e(1i)

has non-zero trace is

1 - p.

As a further

simplification,

we assume the

’probability’

that all elements in

e(1i)

have non-zero

_trace,

to be

_equal

to

_{(1 - p)#~(a).}

Note that these

_assumptions

are _wrong for k = 1. In this _case, the

_trace-map

is the

identity,

and none of

the elements in =

F*

has zero trace. With the above

_assumptions

and the

_equivalence

_(4),

we conclude that the

’probability’

that a

_prime

_p

does not divide a fixed linear _{recurrent sequence} is

approximated

by

In order to obtain a set of

_primes

of

_positive

_density

that do divide the _sequence, we need a set of

positive

density, consisting

of

_primes

_p for

which this

_probability

is

_’small’,

_or,

_{equivalently,}

for which the order of 1i E is

_’large’.

For an

_integer

a ~

_0,

the order of a E

_F*

divides p - 1. In

1927,

Emil Artin

_conjectured

that if a is not

_equal

to -1 or a _square, the set of

_primes

p for which the order of a E

F*

equals

p - 1 has

positive

density.

More

generally,

we

_expect

that for each h E

Z&#x3E;i ,

the set

_Th

of

_primes

_p for which the order of a E F* is

_(p-1)/h,

has a

_density

b(Th).

Note that

8(Th)

is not

positive

for all a and h E

_Z&#x3E;l;

if a is a _square, there are no odd

_primes

_p

for which a E

_F*

has order p - 1.

However,

we do

_expect

the

_equality

8(Th)

= 1 to hold. In other

words,

_we

_expect

that for ’most’

primes

p, the order of a modulo _p is ’almost’ maximal. These

conjectures

are

still _open, but have been

_proved

under the

_assumption

of the

_generalized

Riemann

_hypothesis

_[5,

_7,

_14].

To

_adapt

Artin’s

_conjecture

to our

situation,

we first need an _upper

bound for the order of a E The trivial _upper bound for this order is the

_exponent

_e(p)

of

_(C~/pC~)*.

The value of

_e(p)

as function of

the

_prime

_p

_depends

on the

_splitting

_type

of the characteristic

_polynomial

f

modulo _p. More

_precisely,

for

_primes

p not

_dividing

the discriminant of

_f,

we have

e(p)

=

lcmd(pd -

1),

with d

ranging

_over the

degrees

of the

irreducible factors of

_f

E

_Fp[X].

_Now,

let T be the set of

_primes

with a fixed

splitting

type.

We _say that _{a, or,}

_{equivalently,}

_{f ,}

satisfies the

_generalized

Artin

_conjecture

for

_primitive

_roots, if the

_following

holds.

For each h E the set

_Th

_of

_primes

_{p E T}

_for

which the order

_of

d E is

_e(p)/h,

has a

density

6(Th).

Moreover,

we have

~h 18(Th) _

8(T) .

Artin’s

_original

_conjecture

excludes the

_integers

~1. In our

_general

_setting,

there are more a’s for which the above

conjecture

does not hold.

However,

we

_expect

the

conjecture

to hold for

’generic’

a. As it is not our

_prime

interest to

_classify

the

_exceptional

_a’s,

we do not

_specify

what we mean

by

‘generic’.

Instead we

_give

an

_example

of a

_non-generic

a. Assume

f

does not factor over

Q

as a

product

of linear

polynomials.

By

Chebotarev’s

density

theorem,

the set T of

_primes

_p for which

_f

does not

_{split completely}

into linear factors modulo _p, has a

positive density.

For

p E T,

the

_exponent

e(p)

is at least

_{p2 -1.}

Now assume in addition that

ak

is

rational,

for some

fixed k E The order of a E is at most

_k(P-1),

so

_Th

is finite for all h E We find that

_¿h=18(Th)

= 0 is

strictly

less than

6(T),

and the above

_conjecture

does not hold.

Now we return to our linear recurrent sequence S of order k &#x3E; 2. Assume

that the associated characteristic

_polynomial

of S satisfies the

_generalized

Artin

_conjecture.

For the

_following,

we

distinguish

_{two cases;} either

f splits

into distinct linear factors modulo p, or not. In the former case we _say that

p is a

_splitting

_prime

or that _p

_splits

completely.

First we take for T the set of

_primes

that

_split

_completely.

The set T

is

_isomorphic

to

_F*,

and has

_exponent

_{e(p) =}

p - 1.

By

assumption

there exists h E such that the set

Th

of

_{primes p E T}

for which a E has order

_{(p - 1 ) /h,}

has a

_positive

_density.

According

to our

heuristics,

the

’probability’

that _p E

_Th

does not divide the sequence is

(1 2013 i)(P")/.

For _{p -} oo this

expression

_{converge to} which

p

lies

_strictly

between 0 and 1.

_Therefore,

we

_expect

that

Th

contains two

subsets of

_positive

lower

_density,

one

_consisting

of

_primes

that do divide

the sequence, and one

_consisting

of

_primes

that do not divide the sequence.

As a _{consequence, it} seems

plausible

that the set of

_primes

in T that divide ,S’ has a

positive

lower

density, strictly

less than 1. Note that

although

the

above formula for the

_{’probability’}

does

_give

us an indication of what to

expect,

it does not

_pretend

to

_give

a

_precise

value for the

density

of the set

of

_primes

in

_Th

that divide ,S’.

_Therefore,

our heuristics do not

_predict

the

existence of the

_density

of the set of

_primes

in T that divide S.

Now,

let T be the set of

_primes

that do not

_{split completely.}

For h E we define

Th

as the set of

primes p E T

for which the order of a E

_{(O/P 0)*}

is

_equal

to

_{e(p) /h.}

As

_f

does not

_{split completely}

modulo

_{p E T,}

we have

e(p) ~

p2 -1.

Using

our heuristic

formula,

we

_expect

the

’probability’

that

p E

_Th

does not divide S to be at most

(1 2013 ) )/.

This

_expression

p

converges

exponentially

to

0,

for p - oo.

Therefore,

for all h E the subset of

_Th

of

_primes

that do not divide the sequence, should have zero

density.

According

to our

_{generalization}

of Artin’s

_conjecture,

the

_density

of

_{Uh5:nTh gets}

_arbitrarily

close to

_6(T),

for n -~ oo. As a

conclusion,

we

expect

the set of

_primes

that do not

_{split completely}

and that do not divide the sequence to have zero

density.

Up

to now, we discussed linear recurrent _sequences of _any order k &#x3E; 2.

The theorem below and the data in section 5

_support

the above heuristics for linear recurrent _sequences of order k &#x3E; 3.

_However,

for a

_specific

lin-ear recurrent sequence S of order 2 with irreducible

polynomial,

the above

heuristics are

proved

to be false. Let ,S’ = be the linear recurrent

sequence with irreducible characterisitic

polynomial f

= X 2 -

5X + 7 and initial values xl - 1 and X2 = 2. Let a and a be the roots of

_f

and de-note

_by

0 the

_ring

of

_integers

of

_Q(a).

_Lagarias

_[6]

showed that a

_prime

p divides ~’ if and

only

if the order of

a/a

E is divisible

_by

3. This condition is

_equivalent

with certain

_spitting

conditions for _p in certain

number fields.

_Using

Chebotarev’s

_density

_theorem,

_Lagarias

showed that the set of

_primes

that do not

_{split completely}

in

_K/Q

and that do not divide ~S’ has

_{positive density. According}

to our

_heuristics,

this set should have zero

density.

Let ,S’ be a linear recurrent sequence of 3. Recall from section

2 that a denotes the residue class of X in the

ring

0 =

d is some fixed

integer.

Define for each h E

_Z&#x3E;i

the set

Th

=

{P :

f

is irreducible modulo _p, and

_{[((O/pO)* : (1i)]}

=

~}.

Theorem. Let S be a linear recurrent _sequence of order k &#x3E;

_3,

and define for each

_{positive integer}

h the set

Th

as above. For _every

positive integer

h,

all but

_finitely

_many

_primes

in

_Th

divide S.

We will _prove this theorem in the next section. The main idea of the

_proof

is to relate the

_primes

_p E

Th

that divide S to the existence of

_Fp-rational

points

on some

_projective

_algebraic

_variety

defined over

_Fp.

For

_large

_p,

the existence of these

_points

is

_{guaranteed by}

a result of

Lang

and Weil.

The theorem does not

_imply

that

_Ps

contains a set of

_positive

_density.

There are two

problems.

First of all the set T of

primes p

for

which f

E

Fp[X]

is irreducible

_might

be

_empty,

for

_example

if

_f

is reducible over

_Q.

If

f

is

_irreducible,

a _necessary and sufficient condition for T to have

positive

density,

is the existence of a

_k-cycle

in the Galois _group of

f . Namely,

suppose

f

is irreducible modulo the

prime

p. The

decomposition

group of

a

_prime

above _p in a normal closure of

Q[X]/(f)

acts

_transitively

on the roots of

_{f ,}

and therefore contains a

_k-cycle.

The fact that the existence of a

k-cycle

is sufficient for T to have

_positive

_density

follows from Chebotarev’s

density

theorem.

Much more difficult is the

_question

whether there exists h E

_Z&#x3E;i

such that

Th

has

_positive

_density.

An affirmative answer to this

_question

is

pre-cisely

the

_{generalization}

of Artin’s

_conjecture

that we discussed above. In

order to conclude that

_Ps

has

_positive

lower

_density,

we need the

following

precise

version of this

_conjecture.

Generalized Artin

_conjecture:

Let

_f

E

_Z[X]

be an irreducible monic

polynomial,

a a root of

_f,

and define K =

Q (a)

with

_ring

of

_{integers O K .}

Furthermore,

let T be the set of

_primes

that are inert in and define

for each h E

_Z&#x3E;l,

the set

_Th

of

_{primes p E T}

for which the

_subgroup

(1i)

C has index h.

Then

Th

has a natural

density

6(Th)

for all h E

_Moreover,

we have

the

_equality

=

8(T).

As we saw

before,

_although

it

_happens

that the

_conjecture

fails to hold for

specific

a, we

_expect

the

_conjecture

to hold for

_{’generic’}

a. Based on

com-puter

calculations for

_quadratic

_polynomials,

Brown and Zassenhaus made

a similar

_conjecture

_[2].

Under some mild

assumptions

on the

polynomial

f , they conjectured

that the set

_Tl

has

_positive

_density.

The

_generalized

Artin

_conjecture

has been

_proved

for

_integers

_{a 0}

_{(0, + 1 ) ,}

or,

equivalently,

for linear

polynomials f ,

under the

_assumption

of the

proved

[12]

that

_part

of the

_conjecture

holds for irreducible

_quadratic

poly-nomials

_{f .}

_{Unfortunately,}

we are not able _{to prove} the

_generalized

Artin

conjecture

for a

_{single polynomial}

of

_degree

at least

_3,

not even under the

assumption

of the

_generalized

Riemann

_hypothesis.

Corollary.

Let S be a linear _{recurrent sequence} of

order k &#x3E;

3. Assume

the associated characteristic

_polynomial

satisfies the

_generalized

Artin

con-jecture,

and contains a

_k-cycle

in its Galois _group. Then

_Ps

contains a set of

_positive

_density

Proof.

Let a be a root of the characteristic

_{polynomial f ,}

and let

_OK

be the

_ring

of

_integers

of the number field K =

Q(a).

For all

primes

_{p not}

dividing

d,

the

_ring

_O/pO

is

_isomorphic

to

_By

the

_assumption

on

f ,

we conclude that

Th

has a

density

6(Th)

for all

positive integers

h.

According

to the

_theorem,

the

_density

of the set

_{TS h}

of

_primes

_p E

_Th

that divide S is

_equal

to

_{6(Th) .}

Denote the _upper

_density

and lower

_density

of

Ts

=

T fl PS

by

S(TS)

and

_6(TS)

_{respectively.}

_Using

the above

_remarks,

we

find:

r’V"’B I"V’B

We conclude that the set

_Ts

has a

density

and moreover we have

6(T) .

As the Galois _group of

_f

contains a

k-cycle,

this

density

is

_positive.

D

4. Proof of the theorem

As in section

_2,

we choose an

_integer

d #

0 such that _xn =

where 0 =

R (X ~ / ( f )

is _a free

algebra

_over R =

Z[~j,

and _{c e} O is

uniquely

determined

_by

Fix h E

Z&#x3E;i

and define the set

Let p

E

Th

be a

_prime

not

_dividing

d and write 1i for the reduction of a

modulo

_pO.

_{By assumption}

_f

is irreducible modulo _p, and =

is a field of order

pk.

As is

cyclic,

the

subgroup generated by

1i

consists of the h-th _{powers in}

_((O/pO)*,

and we find

Together

with the

_equivalence

₍₄₎

in section

2,

this

_yields

the

_following

characterization:

We will show that if p is

large enough,

then the

right

hand side of

(5)

is satisfied. This

_implies

that all but

_finitely

_many

_primes

in

_Th

divide

_,S’,

and the theorem is

_proved.

By extending

scalars,

we find that

O[X,,..., Xk]

is a free

JR[Xi,...,

Xk]-algebra.

Define the

_{homogeneous polynomial}

Note that

_Fh

is

_independent

of the

_prime

_p. The coefficient

of

_Xl

in

_Fh

is the h-th term of the _sequence S. If this coefficient is _zero,

then all

_primes

divide S and the theorem is

_{proved. Therefore,}

we assume

that is _non-zero, so

_Fh

is

_homogeneous

of

_degree

h. Let

_Vh

C

be the

_projective

_algebraic

set defined

_by

_F~.

The reduction

_Vh

of

Vh

modulo _p is

_given

_by

the

_polynomial

For _{j?i,... x~} E

_Fp,

the

_equation

on the

_right

hand side of the

_equivalence

(5)

becomes

_{Fh(xl, ... ,}

_Xk)

=

0,

_{so we} arrive _at

where

_Vh(Fp)

denotes the set of

_Fp-rational

_points

of

_Vh.

In order to prove that

_Vh

is a

non-singular

_{projective variety,}

we let

Wh

c

Pk-1 (FP)

be the

_projective

_algebraic

set defined

_by

If _p

_{f h}

and

_{c ~}

0,

then the

_partial

derivatives of

_Gh

do not vanish

si-multaneously on

and

Wh

is a smooth

_projective

_hypersurface

of dimension k - 2. As k is at least

_3,

the intersection of two different

irreducible

_components

of

_Wh

is

_non-empty

_by

_[3,

Theorem 7.2 on _page

48~.

However,

the

_points

of such an intersection are

_singular

_points

of

Wh.

Because

Wh

is

_smooth,

this _proves that

_Wh

is

_absolutely

irreducible. Note that if k

_{equals 2,}

then

_{Wh (Fp)}

consists of h

_points,

and is not

absolutely

irreducible.

The varieties

_{Wh and Vh}

are

_isomorphic

over

_Namely,

we can

The determinant

_{of 0}

is

_equal

to times the Vandermonde determinant of

_{1iP, ... ,}

As d E

_OlpO

is of

_degree

k over

_Fp,

the determinant

is non-zero

_{and 0}

is an

_isomorphism.

We conclude that for all _p E

_Th

that do not divide

_cdh,

the

_algebraic

set

Vh

is a

smooth,

absolutely

irreducible

hypersurface

in

_Pk-1(Fp)

of

_degree

h.

The result of

_Lang

and Weil

_[8,

Theorem

_1]

_yields

the

_following

estimate for the number of rational

_points

of

_Vh:

there exists a constant

C,

depending

on h and k

_only,

such that for all

_{primes p}

E

_Th

not

_dividing

cdh we have

Note that instead of

_[8]

one can also use the

Weil-conjectures

to obtain

this estimate. We conclude that

_{if p E Th is large enough,}

then

_{Vh (Fp)}

is

not

_empty

and

₍₆₎

_implies

that _p divides ~S’. This finishes the

_proof

of the theorem.

5. Numerical

_examples

In the table

_below,

we have collected some numerical data. For each of

the five _recurrences, there are two recurrent _sequences. For each of these

sequences, we determined how _many of the 5133

_primes

_{up to} 5

-104

divide

the _sequence and ordered them

_by

their

_splitting

_type.

For

_example,

for 853

_primes

_{up to} 5 -

_104,

the

_{polynomial f}

=

X~ - 2X 2

₊ X - 6 _E

Fp[X]

splits

into three linear factors. Out of these 853

_primes,

293 divide the

sequence defined

by f

and the initial conditions xi =

1, x2 -

-2 and

X3 = 5. A ’mixed’

splitting

_{type just}

means that

f

E

_Fp[X]

factors as

the

_product

of a linear an a

_{quadratic polynomial.}

The first _{6 sequences}

have an irreducible characteristic

polynomial

with Galois _group

63,

the

symmetric

group on 3 elements. For the next 2 sequences, the

_polynomial

is irreducible with

_cyclic

Galois _group. The characteristic

_polynomial

for the last two sequences factors over

Q

as a

product

of a linear and a

quadratic

factor.

For each of these _sequences, the conclusions that we drew on heuristic

arguments

in section

_3,

seem to be correct. Almost all

_primes

that do not

_{split completely,}

divide the _sequence, and out of the

_primes

that

_split

completely,

a

_{positive proportion}

does divide the _sequence and a

_positive

proportion

does not.

The first two characteristic

_polynomials

are related. If a is a root of

_f

=

X~-2X~+X-6,

then

_~3

is a root

_{of 9}

=

X 3 - 254X2 - 3311X - 46~56.

In other

_words,

the third and the fourth _sequence are

_subsequences

of the

first and the

_second,

_{respectively.}

_Therefore,

the first _sequence has at least

as _many

_prime

divisors as the third _sequence. The heuristical

_arguments

in section 3

_{explain why}

the first _sequence has

_considerably

more

_splitting

prime

divisors than the third _sequence, and

_why

both _sequences have almost the same number of

_{non-splitting}

_prime

divisors. First we note that as a

and

_fl

define

_isomorphic

number

_fields,

the

_splitting

_type

of

_f

modulo _p is

the same as that

of g

modulo _p. Our heuristical model

predicts

that for _any

linear recurrent sequence S of order k &#x3E;

3,

almost all

primes

that do not

split completely

should divide ,S’. In

_particular,

almost all

_primes

that do not

_{split completely}

divide both the first and the third sequence. Now let p be a

splitting

prime,

so the

_{’probability’}

that _p does not divide either of the

sequences is

positive.

If the order of a E

_{(0/ pO) *}

is not

_relatively

_prime

to

6,

the order =

as

is smaller than that of a.

According

to our

_heuristics,

it is then more

likely

for _{p to} divide the first _sequence, than to divide the third. For

_example,

there are no odd

splitting

_primes

_p for which the order

of

_fl

E

_{(0 /pO) *}

is p - l. On the other

hand,

out of the set of

primes

that

split

completely,

the

_primes

_p for which the order of a E

_{(0 /pO) *}

is

_{p -1,}

have the

_{largest ’probability’}

to divide the first _sequence.

_Therefore,

the number of

_splitting

_primes

_dividing

the first _sequence should be

_larger

than the number of

_splitting

_primes

_dividing

the third _sequence.

We conclude with a few observations. As we can see from the fifth and the

sixth sequence, the number of

splitting

primes

dividing

a linear recurrent

seems to

_dependent

not

_only

on the characteristic

polynomial,

but also on

the initial conditions. This can not be

_{explained by}

our heuristics.

Our theorem is in fact

_empty

for the fifth and sixth _sequence.

_Namely,

let a be a root of f

= X 3 - X2 - X -1,

and

_{let p}

be a

_prime

for which

f

is

irreducible modulo p. As a has norm

_1,

the index

(1i)]

is at least

p-1,

and the set

Th

is finite for all h E This

_argument

also shows that the

_generalized

Artin

_conjecture

does not hold for

_{f .}

_However,

the data

suggest

that our heuristical conclusions on the number of

_prime

divisors

is still correct. This can be

explained by

the

_assumption

that for ’most’

primes

p, the order of a E is still ’as

_large

as

possible’.

To be more

_precise,

if we

_adapt

our heuristics

_{by replacing}

e(p)

by

the

_exponent

e(p)

of the kernel of the norm N :

(C~/pC7)* -~

_Fp,

we can draw the same

conclusions as in section 3. For a

_{generalization}

of Artin’s

_conjecture

for

units in real

_quadratic

_fields,

we refer to

_[11].

Finally

we note that further numerical

experiments

seem to

_suggest

that for the _sequences ,5’ in the

_table,

the

_splitting

_primes

_dividing

,S’ have a

density.

As an

example,

from the table we see that

34%

of the 853

splitting

primes

up to

5 ~ 104

divide the first sequence. We

computed

that out of the

first 853

_splitting

_primes

_greater

than 5 -

_104,

this

_percentage

is

35%.

References

[1] C. _{BALLOT, Density of prime} divisors _of linear recurrent sequences. Mem. of the AMS 551 _(1995).

[2] H. BROWN, H. _ZASSENHAUS, Some empirical observations on _primitive roots. J. Number Theory 3 _{(971),306-309.}

[3] R. HARTSHORNE, Algebraic geometry. Springer-Verlag, New York, 1977.

[4] H. _HASSE, Über die Dichte der Primzahlen p, für die eine _{vorgegebene ganz-rationale} Zahl _{a ~ 0} von _gerader bzw. _{ungerader Ordnung mod.p} ist. Math. Ann. 166 (1966),

19-23.

[5] C. HOOLEY, On Artin’s conjecture. J. Reine Angew. Math. 225 _(1967), 209-220.

[6] J.C. LAGARIAS, The set _{of primes dividing} the Lucas numbers has density 2/3. Pacific J. Math. 118 _(1985), 449-461; Errata Ibid. 162 _(1994), 393-397.

[7] H.W. LENSTRA, JR, On Artin’s conjecture and Euclid’s algorithm in _{global fields.} Inv. Math. 42 _(1977), 201-224.

[8] S. LANG, A. _WEIL, Number of points of varieties in _{finite fields.} Amer. J. Math. 76

(1954), 819-827.

[9] P. MOREE, P. _STEVENHAGEN, Prime divisors of Lucas sequences. Acta Arith. 82 _(1997), 403-410.

[10] G. _PÓLYA, Arithmetische _{Eigenschaften} der _{Reihenentwicklungen} rationaler Funktionen. J. Reine _Angew. Math. 151 _{(1921), 99-100.}

[11] H. ROSKAM, A _{Quadratic analogue of} Artin’s _conjecture on _primitive roots. J. Number Theory 81 _{(2000), 93-109.}

[12] H. ROSKAM, Artin’s Primitive Root Conjecture for Quadratic Fields. Accepted for pub-lication in J. Théor. Nombres Bordeaux.

[13] P.J. STEPHENS, Prime divisors of second order linear recurrences. J. Number Theory 8

(1976), 313-332.

[14] S.S. _{WAGSTAFF, Pseudoprimes} and a _{generalization of} Artin’s _conjecture. Acta Arith. 41 _{(1982), 141-150.}

[15] M. _WARD, Prime divisors _of second order _recurring sequences. Duke Math. J. 21 (1954), 607-614.

[16] M. _WARD, The maximal _prime divisors of linear recurrences. Can. J. Math. 6 _(1954), 455-462 Hans ROSKAM Mathematisch Instituut Universiteit Leiden Postbus _9512, 2300 RA Leiden The Netherlands E-mail : roskamQmath.leidenuniv.nl