generalized fermat, double fermat and newton sequences

http://www.elsevier.com/locate/jnt

Journal of Number Theory 98 (2003) 172–183

Generalized Fermat, double Fermat andNewton sequences

Bau-Sen Du,a Sen-Shan Huang,b and Ming-Chia Lib

a Academia Sinica, Institute of Mathematics, Taipei 115, Taiwanb Department of Mathematics, National Changhua University of Education, Changhua 500, Taiwan

Received 18 October 2001

Communicated by D. Goss

Abstract

In this paper, we discuss the relationship among the generalized Fermat, double Fermat,

and Newton sequences. In particular, we show that every double Fermat sequence is a

generalized Fermat sequence, and the set of generalized Fermat sequences, as well as the set of

double Fermat sequences, is closed under term-by-term multiplication. We also prove that

every Newton sequence is a generalized Fermat sequence and vice versa. Finally, we show that

double Fermat sequences are Newton sequences generated by certain sequences of integers.

An approach of symbolic dynamical systems is used to obtain congruence identities.

r 2002 Elsevier Science (USA). All rights reserved.

MSC: 11B39; 11B50; 37B10

Keywords: Generalized Fermat sequence; Double Fermat sequence; Newton sequence; Mobius inversion

formula; Symbolic dynamics; Liouville’s formula; Waring’s formula; de Polignac’s formula

1. Introduction

First of all, we give the definitions of Fermat, generalized Fermat and doubleFermat sequences.

Definition 1. Let fangNn¼1 be a sequence of integers and be simply denoted by fang:We call fang a generalized Fermat sequence (resp. Fermat sequence) if for every nAN

E-mail addresses: [email protected] (B.-S. Du), [email protected] (S.-S. Huang),

[email protected] (M.-C. Li).

0022-314X/02/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved.

PII: S 0 0 2 2 - 3 1 4 X ( 0 2 ) 0 0 0 2 5 - 2

(resp. for every prime number n),Xdjn

mðdÞand� 0 ðmod nÞ;

where m is the Mobius function, that is, mð1Þ ¼ 1; mðmÞ ¼ ð�1Þk if m is a product of k

distinct prime numbers, and mðmÞ ¼ 0 otherwise.We call fang a double Fermat sequence if the following conditions are satisfied:

1.P

djn; d odd mðdÞand� 0 ðmod 2nÞ for any nAN with an odd prime factor.

2. a2k � 1 ðmod 2kþ1Þ for any kAN,f0g:

Fermat’s little theorem states that fang with a an integer is a Fermat sequence.Many generalized Fermat sequences can be obtained by counting numbers ofperiodic points for maps (see [7–11,16,17] for interval maps and see Section 5 foredge-shift maps). Double Fermat sequences arise naturally from the numbers ofsymmetric periodic points for odd interval maps (refer to [6,8,10]), although thedefinition seems artificial at first sight.

In Section 2, we give equivalent criteria for generalized Fermat and double Fermatsequences (Theorem 3). As applications of it, we can show that (i) double Fermatsequences must be generalized Fermat sequences; (ii) double Fermat sequences canbe characterized as generalized Fermat sequences equipped with some congruenceproperty; (iii) every generalized Fermat sequence and every double Fermat sequenceconsist of infinitely many generalized Fermat subsequences and double Fermatsubsequences respectively; and (iv) the term-by-term product of two generalized(resp. double) Fermat sequences is also a generalized (resp. double) Fermatsequence.

Next, we give the definition of Newton sequences generated by sequences ofintegers, which naturally extends the definition of the usual Newton sequencesgenerated by finitely many integers as in [13].

Definition 2. Given a sequence of integers fcng; the Newton sequence fang generatedby fcng is defined by the Newton identities, namely,

an ¼ c1an�1 þ c2an�2 þ?þ cn�1a1 þ ncn:

If there exists kAN such that cn ¼ 0 for all n4k; then we simply call fang the Newton

sequence generated by finitely many integers ci with 1pipk:

It is known that the Newton sequence fang generated by ci with 1pipk satisfies

an ¼ trðAnÞ; where A is the companion matrix of the polynomial xk � c1xk�1 �c2xk�2 �?� ck�1x � ck and trðAnÞ is the trace of An (refer to [12]). The Newton

sequence generated by the sequence itself is of the form fð�1Þn�1ang; this is also a

simple example of a Newton sequence not generated by finitely many integers.

B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183 173

In Section 3, we prove that every Newton sequence generated by integers is ageneralized Fermat sequence, and vice versa (Theorems 5 and 6). As an applicationto congruence identities in number theory, we show that many well-knownsequences, e.g., the Lucas sequences and the Lucas functions, are generalizedFermat sequences.

In Section 4, we show that every double Fermat sequence is a Newton sequencegenerated by a sequence of integers with the first term odd and all the other termseven, and vice versa (Theorem 9). An application to Waring’s formula in algebraiccombinatorics is also shown.

In Section 5, we give the proof of Theorem 5 by using symbolic dynamics ratherthan other possible ways, e.g., p-adic analysis. The reason we use the approach ofsymbolic dynamics is to further investigate the intimate relation between numbers ofperiodic points in dynamical systems and congruence identities in numbers theory,and also to generalize related results in this direction (cf. [7–11,16,17]). A conciseintroduction of symbolic dynamics is included.

2. Generalized Fermat and double Fermat sequences

In this section, we study properties of generalized Fermat and double Fermatsequences. First, we have criteria for generalized Fermat sequences and doubleFermat sequences.

Theorem 3. Let fang be a sequence of integers. Then

1. fang is a generalized Fermat sequence if and only if for any nAN and for any prime

factor p of n so that pt jj n (i.e., pt j n and ptþ1 [ nÞ for some tAN;

an � anpðmod ptÞ:

2. fang is a double Fermat sequence if and only if for any nAN so that 2s jj n for some

sAN,f0g; for any odd prime factor p of n so that pt jj n for some tAN; and for any

kAN,f0g;

an � anpðmod 2sþ1ptÞ and a2k � 1 ðmod 2kþ1Þ:

Proof. We give the proof of item 1 and omit a similar proof of item 2. Let nAN andp be a prime factor of n; thenX

djnmðdÞan

d¼

Xdjn; p[d

mðdÞandþX

djn; pjdmðdÞan

d

¼X

djn; p[d

mðdÞandþ

Xdjn; p[d

mðdpÞa ndp

¼X

djn; p[d

mðdÞðand� a n

dpÞ

B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183174

and so Xdjn

mðdÞand¼ an � an

pþ

Xdjn; p[d; da1

mðdÞðand� a n

dpÞ:

If pt jj n then one has pt jj nd

in the summation of the right-hand side of the above

equality. Therefore, the ‘‘if ’’ part of item 1 follows immediately and the ‘‘only if ’’part can be easily proved by induction on n: &

The following are more properties about generalized Fermat and double Fermatsequences.

Corollary 4. The following statements hold:

1. Every double Fermat sequence is a generalized Fermat sequence.2. Let fang be a generalized Fermat sequence. Then fang is a double Fermat sequence

if and only if a2sm � 1 ðmod 2sþ1Þ for any sAN,f0g and any odd mAN:3. Let fang be a generalized (resp. double) Fermat sequence, then for any kAN the

sequence fankg is a generalized (resp. double) Fermat sequence.4. The set of all generalized Fermat sequences forms a ring under the term-by-term

addition and the term-by-term multiplication, and the set of all double Fermat

sequences is closed under the term-by-term multiplication.

Proof. Items 1, 3 and 4 are evident as applications of the previous theorem. Weprove item 2. By using item 2 of the previous theorem and induction on m; the ‘‘only

if ’’ follows easily. For the ‘‘if ’’ part, it is sufficient to show that an � anpðmod 2sþ1Þ

for any nAN; any odd prime factor p of n and any sX0 with 2s jj n; because of items 1and 2 of the previous theorem. By applying the hypothesis to m ¼ n

2s and n2sp

resp., one

gets that an � 1 ðmod 2sþ1Þ and anp� 1 ðmod 2sþ1Þ resp., which yield the desired

result. &

The above corollary has extended the results and solved the questions in [10];therein only sequences of nonnegative integers are considered.

3. Equivalence between generalized Fermat and Newton sequences

In this section, we give two theorems which together say that generalized Fermatsequences and Newton sequences are essentially the same. Here we state the first onebut postpone its proof to Section 5.

Theorem 5. Every Newton sequence generated by integers is a generalized Fermat

sequence.


In [4], Dickson showed that every Newton sequence generated by finitely manyintegers with zero as the first generator is a Fermat sequence. In [12], Gillespieextended the result by waving the restriction on the first generator. Theorem 5 is ageneralization of both results and, in addition, a dynamical systems approach,different from theirs, will be presented in Section 5.

Next, we list some well-known sequences which can be interpreted as Newtonsequences generated by integers and so, by Theorem 5, they are generalized Fermatsequences. Some of these results are new.

1. The sequence fang with a an integer is the Newton sequence generated by c1 ¼ a:In this case, Theorem 5 takes the form pjap � a for all prime numbers p; which isthe celebrated Fermat’s little theorem.

2. The k-Lucas sequence fang; defined by an ¼ 1 for 1pnpk � 1; ak ¼ k þ 1; andan ¼ an�1 þ an�k for nXk þ 1; is the Newton sequence generated by c1 ¼ 1; c2 ¼c3 ¼ ? ¼ ck�1 ¼ 0; ck ¼ 1: In particular, the 2-Lucas sequence 1; 3; 4; 7; 11;y is ageneralized Fermat sequence, as given in [21].

3. A special type of the generalized k-Fibonacci sequence fang; defined by an ¼ 2n � 1for 1pnpk and an ¼ an�1 þ an�2 þ?þ an�k for nXk þ 1; is the Newtonsequence generated by c1 ¼ c2 ¼ ? ¼ ck ¼ 1: So it is a generalized Fermat

sequence, as obtained in [7]. Note that akþ1 ¼ ð2kþ1 � 1Þ � ðk þ 1Þ and an ¼2an�1 � an�ðkþ1Þ for all nXk þ 2; and so the sequence fang can also be viewed as

the Newton sequence generated by c1 ¼ 2; c2 ¼ ? ¼ ck ¼ 0 and ckþ1 ¼ �1:

4. The Lucas functions Lnðy; zÞ with nX1 are defined by L1ðy; zÞ ¼ y; L2ðy; zÞ ¼y2 � 2z; and Lnðy; zÞ ¼ yLn�1ðy; zÞ � zLn�2ðy; zÞ for nX3; (refer to Dickson’selaborate book [5, Chapter XVII]). For integers y and z; the sequence fLnðy; zÞg isthe Newton sequence generated by c1 ¼ y and c2 ¼ �z and so is a generalizedFermat sequence. In particular, when y is even, say y ¼ 2m; and z ¼ �1; the

functions PnðmÞ ¼ Lnð2m;�1Þ ¼ ðm þffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 1

pÞn þ ðm �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ 1

pÞn are the so-

called Pell–Lucas polynomials. Another special case when y ¼ 2m is even and

z ¼ 1; the functions TnðmÞ ¼ 12Lnð2m; 1Þ ¼ 1

2ðm þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1

pÞn þ 1

2ðm �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1

pÞn

are called the Tchebycheff polynomials of the first kind and so the sequencef2TnðmÞg is a generalized Fermat sequence; this was stated in [12].

We show that the converse of Theorem 5 is valid.

Theorem 6. Every generalized Fermat sequence is a Newton sequence generated by

integers.

Proof. Let fang be a generalized Fermat sequence. Define the sequence fcng byc1 ¼ a1 and the recurrence relation

cn ¼ an � ðc1an�1 þ c2an�2 þ?þ cn�1a1Þn

:


It is evident that fang is the Newton sequence generated by fcng: We show that allthe cn are integers by induction on n: For n ¼ 1; c1 ¼ a1 is an integer. Assume thateach of c1; c2;y; ck is an integer. Let fbng be the Newton sequence generated by ci

with 1pipk: Then we get recurrently that for 1pmpk; bm ¼ c1am�1 þ c2am�2 þ?þ cm�1a1 þ mcm ¼ am; and so bkþ1 ¼ c1ak þ c2ak�1 þ?þ cka1: The assumptionhere and Theorem 5 imply that both fang and fbng are generalized Fermatsequences. Therefore, k þ 1 divides both

akþ1 þX

djðkþ1Þ; da1

mðdÞakþ1d

and bkþ1 þX

djðkþ1Þ; da1

mðdÞbkþ1d

:

Since am ¼ bm for all 1pmpk; the last two sums are identical and so k þ 1 divides

akþ1 � bkþ1: Therefore, ckþ1 ¼ akþ1�bkþ1

kþ1is an integer. The induction has been

completed and so does the proof of the theorem. &

4. Interrelation between double Fermat and Newton sequences

In this section, we investigate the relationship between double Fermat and Newtonsequences. First, we give an explicit formula for each term of a Newton sequence interms of its generators (cf. [2] and [22]).

Proposition 7. Let fang be the Newton sequence generated by fcng; then for any nAN;

an ¼X

k1þ2k2þ?þnkn¼n

n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;y; kn

!ck1

1 ck2

2 ?cknn ;

where ðk1þk2þ?þkn

k1;k2;y;knÞ ¼ ðk1þk2þ?þknÞ!

k1!k2!?kn!with 0! ¼ 1 by convention and the symbol 00; if it

occurs, is interpreted as the value 1.

Proof. Fix nAN and let A denote the companion matrix of xn � c1xn�1 �?�cn�1x � cn; then an ¼ trðAnÞ: By Liouville’s formula (refer to [20]), we have that

XNm¼1

trðAmÞm

tm ¼ log1

detðI � tAÞ

� �

¼ log1

1 � c1t � c2t2 �?� cntn

� �

¼XNk¼1

1

kðc1t þ c2t2 þ?þ cntnÞk


¼XNk¼1

Xk1þk2þ?þkn¼k

k1 þ k2 þ?þ kn

k1; k2;y; kn

!ck1

1 ck2

2 ?cknn

k1 þ k2 þ?þ kn

tk1þ2k2þ?þnkn

( )

¼XNm¼1

Xk1þ2k2þ?þnkn¼m

k1 þ k2 þ?þ kn

k1; k2;?; kn

!ck1

1 ck2

2 ?cknn

k1 þ k2 þ?þ kn

( )tm:

By equating the coefficient of tn; we obtain that

an ¼ trðAnÞ ¼X

k1þ2k2þ?þnkn¼n

n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;?; kn

!ck1

1 ck2

2 ?cknn :

Since nAN is arbitrary, the proof of the proposition is completed. &

The following was pointed out by Peter J.-S. Shiue and it indicates that the aboveproposition is relevant to the so-called Waring’s formula in the theory of algebraiccombinatorics [2,3,15].

Remark 8. Let en; hn and pn be elementary, homogeneous, and power sum symmetric

functions in variables xi with 1pipn; respectively. It is known that nen ¼Pni¼1 ð�1Þi�1

pien�i and nhn ¼Pn

i¼1 pihn�i (refer to [1,14]). By the above proposition,

we get that the following two expressions are both equal to pn in terms of ei’s andhi’s, respectively:

Xk1þ2k2þ?þnkn¼n

ð�1Þk2þk4þ?þk

2½n2� n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;y; kn

!ek1

1 ek2

2 ?eknn

and

Xk1þ2k2þ?þnkn¼n

ð�1Þk1þk2þ?þkn�1n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;?; kn

!hk1

1 hk2

2 ?hknn :

Additionally, in the third example below Theorem 5, we have that ci ¼ 1 with1pipn generates the Newton sequence having the nth term 2n � 1; and so the above


proposition implies that

Xk1þ2k2þ?þnkn¼n

n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;y; kn

!¼ 2n � 1:

The following theorem characterizes a double Fermat sequence as a Newtonsequence with an additional restriction on the generators.

Theorem 9. Every double Fermat sequence is a Newton sequence generated by fcngwith c1 odd and cn even for n41; and vice versa.

Proof. By Theorem 3 and item 2 of Corollary 4, all we need is to show that for theNewton sequence fang generated by fcng; the following two statements areequivalent:

(S1) a2km � 1 ðmod 2kþ1Þ for any kX0 and odd mX1:(S2) c1 is odd and all the other cn are even.

First, we prove (S2) implies (S1). Since a1 ¼ c1 is odd, the result is valid for k ¼ 0

and m ¼ 1: Let n ¼ 2km41 with m odd. By the previous proposition, we have

an ¼ cn1 þ

Xk1þ2k2þ?þnkn¼n; k1on

2km

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;y; kn

!ck1

1 ck2

2 ?cknn :

It is easy to see that c2km1 � 1 ðmod 2kþ1Þ since c1 is odd. Then, it suffices to show that

each term in the last summation is divisible by 2kþ1: Let b denote the largest integer

such that 2b divides 2kmk1þk2þ?þkn

ðk1þk2þ?þkn

k1;k2;?;knÞck1

1 ck2

2 ?cknn : Then, it remains to show that

bXk þ 1 all the time. By the assumption on parity of cn and de Polignac’s formulafrom elementary number theory [19], we have

bX k þXNj¼1

k1 þ k2 þ?þ kn � 1

2j

��XNj¼1

k1

2j

� !

þ k2 �XNj¼1

k2

2j

� !þ k3 �

XNj¼1

k3

2j

� !þ?þ kn �

XNj¼1

kn

2j

� !

p k þ k2 �XNj¼1

k2

2j

!þ k3 �

XNj¼1

k3

2j

!þ?þ kn �

XNj¼1

kn

2j

!4k;


where ½x� denotes the greatest integer less than or equal to x and the last inequalityholds since k1on and then there exists some ki with i41 such that ki40:

Next, we show (S1) implies (S2). Since c1 ¼ a1 � 1 ðmod 2Þ; c1 is odd. Suppose

that the desired result is false. Then there is a smallest n ¼ 2km41 with kX0 and oddmX1 such that cn is odd. By the previous proposition, we obtain that

ncn ¼ an � cn1 �

Xk1þ2k2þ?þnkn¼n; k1on; kn¼0

n

k1 þ k2 þ?þ kn

k1 þ k2 þ?þ kn

k1; k2;y; kn

!ck1

1 ck2

2 ?cknn :

In the last equality, the term ncnð¼ 2kmcnÞ is divisible by 2k; but not divisible by 2kþ1:

Moreover, an � cn1 � 0 ðmod 2kþ1Þ since an � 1 ðmod 2kþ1Þ and c1 is odd. But by

using de Polignac’s formula as above, we have that the last summation is divisible by

2kþ1; a contradiction. Hence, all the cn with n41 are even.We have finished the proof of the theorem. &

5. Proof of Theorem 5: an approach of symbolic dynamics

In this section, we use the theory of symbolic dynamical systems to proveTheorem 5.

First, we give basic definitions in symbolic dynamics, refer to [18,20]. A graph G

consists of a finite set S of states together with a finite set E of edges. Each edge eAE

has initial state iðeÞ and terminal state tðeÞ: Let A ¼ ½AIJ � be a k k matrix withnonnegative integer entries. The graph of A is the graph GA with state set S ¼f1; 2;y; kg and with AIJ distinct edges from edge set E with initial state I andterminal state J: The edge shift space SA is the space of sequences of edges from E

specified by

SA ¼ fe0e1e2?jeiAE and tðeiÞ ¼ iðeiþ1Þ for all integers iX0g:

The shift map sA : SA-SA is defined to be

sAðe0e1e2e3?Þ ¼ e1e2e3?:

For nAN; let snA denote the composition of sA with itself n times. A point

%e ¼

e0e1e2?ASA is called a period-n point for sA if snAð%eÞ ¼

%e and sj

Að%eÞa

%e for 1pjpn �

1: Let PernðsAÞ denote the set of all period-n points for sA and let #PernðsAÞ denotethe cardinal number of the set PernðsAÞ:

It is clear that #PernðsAÞ is finite and divisible by n; and moreover, #Per1ðsnAÞ ¼P

djn #PerdðsAÞ: According to the Mobius inversion formula or the inclusion–


exclusion principle (refer to [19]), we have #PernðsAÞ ¼P

djn mðdÞ#Per1ðsndAÞ: Thus,

the sequence f#Per1ðsnAÞg is a generalized Fermat sequence.

On the other hand, the fact that trðAnÞ ¼ #Per1ðsnAÞ can be easily proved as

follows. Let S be the state set and let E denote the edge set from the graph of A: Afinite sequence of edges from E; denoted by p ¼ e0e1?em; is called a path of length

m þ 1 from iðe0Þ to tðemÞ if tðeiÞ ¼ iðeiþ1Þ for 0pipm � 1: For nAN; let ðAnÞIJ be the

ðI ; JÞth entry of An and let Pðn; I ; JÞ be the number of paths of length n þ 1 from I toJ: By induction on n; one can show that for every nAN; ðAnÞIJ ¼ Pðn; I ; JÞ for all

choices of states I and J in S: Hence,

trðAnÞ ¼XIAS

ðAnÞII ¼XIAS

Pðn; I ; IÞ ¼ #Per1ðsnAÞ:

So, we have shown the following lemma.

Lemma 10. Let A be a square matrix with nonnegative integer entries, then the

sequence ftrðAnÞg is a generalized Fermat sequence.

Next, we consider integral matrices with possibly negative entries. For a matrixA ¼ ½AIJ � with integer entries, let jAj ¼ ½jAjIJ � denote the corresponding matrix of

absolute values so that jAjIJ ¼ jAIJ j; and let Aþ and A� denote the positive and

negative parts of A so that Aþ and A� are the unique matrices with nonnegative

entries satisfying A ¼ Aþ � A� and jAj ¼ Aþ þ A�: It follows that trðAnÞ þ

trðjAjnÞ ¼ trA 0

A� jAj

�n� �: On the other hand, the two matrices

A 0A� jAj

�and

Aþ A�

A� Aþ

�are similar and so the traces of their nth powers are equal, because

A 0A� jAj

�¼ I �I

0 I

�Aþ A�

A� Aþ

�I I

0 I

�where Id is the identity matrix with the

same dimension as A: Thus

trðAnÞ ¼ trAþ A�

A� Aþ

" #n !� trðjAjnÞ:

Therefore, the previous lemma implies the following one.

Lemma 11. Let A be an integral square matrix with possibly negative entries, then the

sequence ftrðAnÞg is a generalized Fermat sequence.

Finally, we are in the position to prove Theorem 5 by applying Lemma 11 to acompanion matrix.


Proof of Theorem 5. Let fang be the Newton sequence generated by fcng: Let kX1and let fbng be the Newton sequence generated by ci with 1pipk: Then bn ¼ an for

all 1pnpk: Let A denote the companion matrix of xk � c1xk�1 �?� ck�1x � ck;then bn ¼ trðAnÞ for all nAN: By the previous lemma, fbng is a generalized Fermatsequence. Therefore, for 1pnpk;X

djnmðdÞan

d¼Xdjn

mðdÞbnd

is divisible by n:

Since kX1 is arbitrary, we have that fang is a generalized Fermat sequence. &

Acknowledgments

The authors thank Professor Peter Jau-Shyong Shiue for invaluable suggestionswhich led to improvements in this paper.

References

[1] P.J. Cameron, Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press,

Cambridge, 1994.

[2] W.Y.C. Chen, The combinatorial power of the companion matrix, Linear Algebra and Its

Applications 232 (1996) 261–278.

[3] W.Y.C. Chen, K.-W. Lih, Y.-N. Yeh, Cyclic tableaux and symmetric functions, Stud. Appl. Math. 94

(1995) 327–339.

[4] L.E. Dickson, Solution of problem #151, Amer. Math. Monthly 15 (1908) 209.

[5] L.E. Dickson, History of the Theory of Numbers, Vol. I: Divisibility and Primality, Chelsea

Publishing Co., New York, 1966.

[6] B.-S. Du, Symmetric periodic orbits of continuous odd functions on the interval, Bull. Inst. Math.

Acad. Sinica 16 (1988) 1–48.

[7] B.-S. Du, A simple method which generates infinitely many congruence identities, Fibonacci Quart.

27 (1989) 116–124.

[8] B.-S. Du, Congruence identities arising from dynamical systems, Appl. Math. Lett. 12 (1999)

115–119.

[9] B.-S. Du, The linearizations of cyclic permutations have rational zeta functions, Bull. Austral. Math.

Soc. 62 (2000) 287–295.

[10] B.-S. Du, Obtaining new dividing formulas njQðnÞ from the known ones, Fibonacci Quart. 38 (2000)

217–222.

[11] M. Frame, B. Johnson, J. Sauerberg, Fixed points and Fermat: a dynamical systems approach to

number theory, Amer. Math. Monthly 107 (2000) 422–428.

[12] F.S. Gillespie, A generalization of Fermat little theorem, Fibonacci Quart. 27 (1989) 109–115.

[13] R.A. Horn, C.A. Johnson, Matrix Analysis, (corrected reprint of the 1985 original Ed.), Cambridge

University Press, Cambridge, 1990.

[14] L.C. Hsu, P.J.-S. Shiue, Cycle indicators and special functions, Ann. Combin. 5 (2001) 179–196.

[15] R. Lidl, H. Niederreiter, Introduction to Finite Fields and Their Applications, (Revision of the 1986

first Ed.), Cambridge University Press, Cambridge, 1994.

[16] C.-L. Lin, Obtaining dividing formulas njQðnÞ from iterated maps, Fibonacci Quart. 36 (1998)

118–124.


[17] C.-L. Lin, A unified way for obtaining dividing formula njQðnÞ; Taiwanese J. Math. 2 (1998) 469–481.

[18] D. Lind, B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University

Press, Cambridge, 1995.

[19] I. Niven, H.S. Zuckerman, H.L. Montgomery, An Introduction to the Theory of Numbers, 5th

Edition, Wiley, New York, 1991.

[20] C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, 2nd Edition, CRC

Press, Boca Raton, FL, 1999.

[21] A. Rotkiewicz, Problems on Fibonacci numbers and their generalizations, in: G.E. Bergum, A.N.

Philippou, A.F Horadam (Eds.), Fibonacci Numbers and Their Applications, D. Reidel Publishing

Company, Dordrecht, Holland, 1986, pp. 241–255.

[22] Z.-H. Sun, Linear recursive sequences and powers of matrices, Fibonacci Quart. 39 (2001) 339–351.


generalized fermat, double fermat and newton sequences

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