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Generalized Fermat, double Fermat and Newton sequences
Journal of Number Theory 98 (2003) 172–183
http://www.elsevier.com/locate/jnt
Generalized Fermat, double Fermat and Newton sequences Bau-Sen Du,a Sen-Shan Huang,b and Ming-Chia Lib b
a Academia Sinica, Institute of Mathematics, Taipei 115, Taiwan 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; Mo¨bius 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 double Fermat sequences. Definition 1. Let fan gN n¼1 be a sequence of integers and be simply denoted by fan g: We call fan g 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
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(resp. for every prime number n), X djn
173
mðdÞan 0 ðmod nÞ; d
where m is the Mo¨bius function, that is, mð1Þ ¼ 1; mðmÞ ¼ ð1Þk if m is a product of k distinct prime numbers, and mðmÞ ¼ 0 otherwise. We call fan g a double Fermat sequence if the following conditions are satisfied: P 1. djn; d odd mðdÞan 0 ðmod 2nÞ for any nAN with an odd prime factor. d
2. a2k 1 ðmod 2kþ1 Þ for any kAN,f0g:
Fermat’s little theorem states that fan g with a an integer is a Fermat sequence. Many generalized Fermat sequences can be obtained by counting numbers of periodic points for maps (see [7–11,16,17] for interval maps and see Section 5 for edge-shift maps). Double Fermat sequences arise naturally from the numbers of symmetric periodic points for odd interval maps (refer to [6,8,10]), although the definition seems artificial at first sight. In Section 2, we give equivalent criteria for generalized Fermat and double Fermat sequences (Theorem 3). As applications of it, we can show that (i) double Fermat sequences must be generalized Fermat sequences; (ii) double Fermat sequences can be characterized as generalized Fermat sequences equipped with some congruence property; (iii) every generalized Fermat sequence and every double Fermat sequence consist of infinitely many generalized Fermat subsequences and double Fermat subsequences respectively; and (iv) the term-by-term product of two generalized (resp. double) Fermat sequences is also a generalized (resp. double) Fermat sequence. Next, we give the definition of Newton sequences generated by sequences of integers, which naturally extends the definition of the usual Newton sequences generated by finitely many integers as in [13]. Definition 2. Given a sequence of integers fcn g; the Newton sequence fan g generated by fcn g is defined by the Newton identities, namely, an ¼ c1 an1 þ c2 an2 þ ? þ cn1 a1 þ ncn : If there exists kAN such that cn ¼ 0 for all n4k; then we simply call fan g the Newton sequence generated by finitely many integers ci with 1pipk: It is known that the Newton sequence fan g generated by ci with 1pipk satisfies an ¼ trðAn Þ; where A is the companion matrix of the polynomial xk c1 xk1 c2 xk2 ? ck1 x ck and trðAn Þ is the trace of An (refer to [12]). The Newton sequence generated by the sequence itself is of the form fð1Þn1 an g; this is also a simple example of a Newton sequence not generated by finitely many integers.
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In Section 3, we prove that every Newton sequence generated by integers is a generalized Fermat sequence, and vice versa (Theorems 5 and 6). As an application to congruence identities in number theory, we show that many well-known sequences, e.g., the Lucas sequences and the Lucas functions, are generalized Fermat sequences. In Section 4, we show that every double Fermat sequence is a Newton sequence generated by a sequence of integers with the first term odd and all the other terms even, and vice versa (Theorem 9). An application to Waring’s formula in algebraic combinatorics is also shown. In Section 5, we give the proof of Theorem 5 by using symbolic dynamics rather than other possible ways, e.g., p-adic analysis. The reason we use the approach of symbolic dynamics is to further investigate the intimate relation between numbers of periodic 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 concise introduction of symbolic dynamics is included. 2. Generalized Fermat and double Fermat sequences In this section, we study properties of generalized Fermat and double Fermat sequences. First, we have criteria for generalized Fermat sequences and double Fermat sequences. Theorem 3. Let fan g be a sequence of integers. Then 1. fan g 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 an ðmod pt Þ: p
2. fan g 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 an ðmod 2sþ1 pt Þ p
a2k 1 ðmod 2kþ1 Þ:
and
Proof. We give the proof of item 1 and omit a similar proof of item 2. Let nAN and p be a prime factor of n; then X X X mðdÞan ¼ mðdÞan þ mðdÞan djn
d
djn; p[d
¼
X
djn; p[d
¼
X
djn; p[d
d
djn; pjd
mðdÞan þ d
X
djn; p[d
mðdÞðan a n Þ d
dp
d
mðdpÞa n
dp
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175
and so X djn
X
mðdÞan ¼ an an þ d
p
djn; p[d; da1
mðdÞðan a n Þ: d
dp
If pt jj n then one has pt jj dn 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 Fermat sequences. Corollary 4. The following statements hold: 1. Every double Fermat sequence is a generalized Fermat sequence. 2. Let fan g be a generalized Fermat sequence. Then fan g is a double Fermat sequence if and only if a2s m 1 ðmod 2sþ1 Þ for any sAN,f0g and any odd mAN: 3. Let fan g be a generalized (resp. double) Fermat sequence, then for any kAN the sequence fank g 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. We prove 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 an ðmod 2sþ1 Þ p
for any nAN; any odd prime factor p of n and any sX0 with 2s jj n; because of items 1 and 2 of the previous theorem. By applying the hypothesis to m ¼ 2ns and 2ns p resp., one gets that an 1 ðmod 2sþ1 Þ and an 1 ðmod 2sþ1 Þ resp., which yield the desired p
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 Fermat sequences and Newton sequences are essentially the same. Here we state the first one but postpone its proof to Section 5. Theorem 5. Every Newton sequence generated by integers is a generalized Fermat sequence.
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In [4], Dickson showed that every Newton sequence generated by finitely many integers with zero as the first generator is a Fermat sequence. In [12], Gillespie extended the result by waving the restriction on the first generator. Theorem 5 is a generalization 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 Newton sequences generated by integers and so, by Theorem 5, they are generalized Fermat sequences. Some of these results are new. 1. The sequence fan g 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 is the celebrated Fermat’s little theorem. 2. The k-Lucas sequence fan g; defined by an ¼ 1 for 1pnpk 1; ak ¼ k þ 1; and an ¼ an1 þ ank for nXk þ 1; is the Newton sequence generated by c1 ¼ 1; c2 ¼ c3 ¼ ? ¼ ck1 ¼ 0; ck ¼ 1: In particular, the 2-Lucas sequence 1; 3; 4; 7; 11; y is a generalized Fermat sequence, as given in [21]. 3. A special type of the generalized k-Fibonacci sequence fan g; defined by an ¼ 2n 1 for 1pnpk and an ¼ an1 þ an2 þ ? þ ank for nXk þ 1; is the Newton sequence 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 ¼ 2an1 anðkþ1Þ for all nXk þ 2; and so the sequence fan g 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 Ln ðy; zÞ with nX1 are defined by L1 ðy; zÞ ¼ y; L2 ðy; zÞ ¼ y2 2z; and Ln ðy; zÞ ¼ yLn1 ðy; zÞ zLn2 ðy; zÞ for nX3; (refer to Dickson’s elaborate book [5, Chapter XVII]). For integers y and z; the sequence fLn ðy; zÞg is the Newton sequence generated by c1 ¼ y and c2 ¼ z and so is a generalized Fermat sequence. In particular, when y is even, say y ¼ 2m; and z ¼ 1; the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi functions Pn ðmÞ ¼ Ln ð2m; 1Þ ¼ ðm þ m2 þ 1Þn þ ðm m2 þ 1Þn are the socalled Pell–Lucas polynomials. Another special case when y ¼ 2m is even and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ 1; the functions Tn ðmÞ ¼ 12Ln ð2m; 1Þ ¼ 12ðm þ m2 1Þn þ 12ðm m2 1Þn are called the Tchebycheff polynomials of the first kind and so the sequence f2Tn ðmÞg 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 fan g be a generalized Fermat sequence. Define the sequence fcn g by c1 ¼ a1 and the recurrence relation cn ¼
an ðc1 an1 þ c2 an2 þ ? þ cn1 a1 Þ : n
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177
It is evident that fan g is the Newton sequence generated by fcn g: We show that all the cn are integers by induction on n: For n ¼ 1; c1 ¼ a1 is an integer. Assume that each of c1 ; c2 ; y; ck is an integer. Let fbn g be the Newton sequence generated by ci with 1pipk: Then we get recurrently that for 1pmpk; bm ¼ c1 am1 þ c2 am2 þ ? þ cm1 a1 þ mcm ¼ am ; and so bkþ1 ¼ c1 ak þ c2 ak1 þ ? þ ck a1 : The assumption here and Theorem 5 imply that both fan g and fbn g are generalized Fermat sequences. Therefore, k þ 1 divides both X
akþ1 þ
mðdÞakþ1
djðkþ1Þ; da1
d
and
bkþ1 þ
X
mðdÞbkþ1 :
djðkþ1Þ; da1
d
Since am ¼ bm for all 1pmpk; the last two sums are identical and so k þ 1 divides bkþ1 is an integer. The induction has been akþ1 bkþ1 : Therefore, ckþ1 ¼ akþ1kþ1 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 Newton sequences. First, we give an explicit formula for each term of a Newton sequence in terms of its generators (cf. [2] and [22]). Proposition 7. Let fan g be the Newton sequence generated by fcn g; then for any nAN;
an ¼
X k1 þ2k2 þ?þnkn ¼n
! k1 þ k2 þ ? þ kn k1 k2 n c1 c2 ?cknn ; k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
2 þ?þkn Þ! 2 þ?þkn Þ ¼ ðk1kþk with 0! ¼ 1 by convention and the symbol 00 ; if it where ðk1kþk 1 ;k2 ;y;kn 1 !k2 !?kn ! occurs, is interpreted as the value 1.
Proof. Fix nAN and let A denote the companion matrix of xn c1 xn1 ? cn1 x cn ; then an ¼ trðAn Þ: By Liouville’s formula (refer to [20]), we have that N X trðAm Þ m 1 t ¼ log m detðI tAÞ m¼1 1 ¼ log 1 c1 t c2 t 2 ? cn t n N X 1 ðc1 t þ c2 t2 þ ? þ cn tn Þk ¼ k k¼1
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! ) k1 þ k2 þ ? þ kn ck11 ck22 ?cknn ¼ tk1 þ2k2 þ?þnkn k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn k¼1 k1 þk2 þ?þkn ¼k ( ! ) N X X k1 þ k2 þ ? þ kn ck11 ck22 ?cknn ¼ tm : k þ k þ ? þ k k ; k ; ?; k 1 2 n 1 2 n m¼1 k þ2k þ?þnk ¼m N X
(
X
1
n
2
By equating the coefficient of tn ; we obtain that ! k1 þ k2 þ ? þ kn k1 k2 n an ¼ trðA Þ ¼ c1 c2 ?cknn : k þ k þ ? þ k k ; k ; ?; k 2 n 1 2 n k1 þ2k2 þ?þnkn ¼n 1 X
n
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 above proposition is relevant to the so-called Waring’s formula in the theory of algebraic combinatorics [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 ¼ P Pn i1 pi eni and nhn ¼ ni¼1 pi hni (refer to [1,14]). By the above proposition, i¼1 ð1Þ we get that the following two expressions are both equal to pn in terms of ei ’s and hi ’s, respectively: X
ð1Þ
k2 þk4 þ?þk n 2½ 2
k1 þ2k2 þ?þnkn ¼n
k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
!
n k1 þ k2 þ ? þ kn
ek11 ek22 ?eknn
and X k1 þ2k2 þ?þnkn ¼n
ð1Þk1 þk2 þ?þkn1
n k1 þ k2 þ ? þ kn
! k1 þ k2 þ ? þ kn k1 k2 h1 h2 ?hknn : k1 ; k2 ; ?; kn
Additionally, in the third example below Theorem 5, we have that ci ¼ 1 with 1pipn generates the Newton sequence having the nth term 2n 1; and so the above
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proposition implies that X k1 þ2k2 þ?þnkn ¼n
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 Newton sequence with an additional restriction on the generators. Theorem 9. Every double Fermat sequence is a Newton sequence generated by fcn g with 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 the Newton sequence fan g generated by fcn g; the following two statements are equivalent: (S1) a2k m 1 ðmod 2kþ1 Þ 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 2k m41 with m odd. By the previous proposition, we have
k
X
2k m k þ k2 þ ? þ kn k1 þ2k2 þ?þnkn ¼n; k1 on 1 ! k1 þ k2 þ ? þ kn k1 k2 c1 c2 ?cknn : k1 ; k2 ; y; kn
an ¼ cn1 þ
It is easy to see that c21 m 1 ðmod 2kþ1 Þ 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 k m 2 þ?þkn such that 2b divides k1 þk22þ?þk ðk1kþk Þck11 ck22 ?cknn : Then, it remains to show that n 1 ;k2 ;?;kn bXk þ 1 all the time. By the assumption on parity of cn and de Polignac’s formula from elementary number theory [19], we have ! X N
N X k1 þ k2 þ ? þ kn 1 k1 bX k þ 2j 2j j¼1 j¼1 ! ! ! N N N X X X k2 k3 kn þ k2 þ k3 þ ? þ kn 2j 2j 2j j¼1 j¼1 j¼1 ! ! ! N N N X X X k2 k3 kn p k þ k2 þ k þ ? þ k 4k; 3 n j j 2 2 2j j¼1 j¼1 j¼1
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where ½x denotes the greatest integer less than or equal to x and the last inequality holds since k1 on and then there exists some ki with i41 such that ki 40: 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 2k m41 with kX0 and odd mX1 such that cn is odd. By the previous proposition, we obtain that X
ncn ¼ an cn1
k1 þ2k2 þ?þnkn ¼n; k1 on;
k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
!
n k þ k2 þ ? þ kn k ¼0 1 n
ck11 ck22 ?cknn :
In the last equality, the term ncn ð¼ 2k mcn Þ is divisible by 2k ; but not divisible by 2kþ1 : Moreover, an cn1 0 ðmod 2kþ1 Þ since an 1 ðmod 2kþ1 Þ 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 prove Theorem 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 with nonnegative integer entries. The graph of A is the graph GA with state set S ¼ f1; 2; y; kg and with AIJ distinct edges from edge set E with initial state I and terminal state J: The edge shift space SA is the space of sequences of edges from E specified by SA ¼ fe0 e1 e2 ?jei AE and tðei Þ ¼ iðeiþ1 Þ for all integers iX0g: The shift map sA : SA -SA is defined to be sA ðe0 e1 e2 e3 ?Þ ¼ e1 e2 e3 ?: For nAN; let snA denote the composition of sA with itself n times. A point e ¼ % e0 e1 e2 ?ASA is called a period-n point for sA if snA ðeÞ ¼ e and sjA ðeÞae for 1pjpn % ðs Þ denote 1: Let Pern ðsA Þ denote the set of all period-n points% for %sA and let% #Per n A the cardinal number of the set Pern ðsA Þ: It is clear that #Pern ðsA Þ is finite and divisible by n; and moreover, #Per1 ðsnA Þ ¼ P djn #Perd ðsA Þ: According to the Mo¨bius inversion formula or the inclusion–
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181
n P exclusion principle (refer to [19]), we have #Pern ðsA Þ ¼ djn mðdÞ#Per1 ðsdA Þ: Thus, the sequence f#Per1 ðsnA Þg is a generalized Fermat sequence. On the other hand, the fact that trðAn Þ ¼ #Per1 ðsnA Þ can be easily proved as follows. Let S be the state set and let E denote the edge set from the graph of A: A finite sequence of edges from E; denoted by p ¼ e0 e1 ?em ; is called a path of length m þ 1 from iðe0 Þ to tðem Þ if tðei Þ ¼ iðeiþ1 Þ for 0pipm 1: For nAN; let ðAn ÞIJ be the ðI; JÞth entry of An and let Pðn; I; JÞ be the number of paths of length n þ 1 from I to J: By induction on n; one can show that for every nAN; ðAn ÞIJ ¼ Pðn; I; JÞ for all choices of states I and J in S: Hence,
trðAn Þ ¼
X
ðAn ÞII ¼
IAS
X
Pðn; I; IÞ ¼ #Per1 ðsnA Þ:
IAS
So, we have shown the following lemma. Lemma 10. Let A be a square matrix with nonnegative integer entries, then the sequence ftrðAn Þg is a generalized Fermat sequence. Next, we consider integral matrices with possibly negative entries. For a matrix A 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 trðAn Þ þ n
A 0 A 0 n trðjAj Þ ¼ tr and : On the other hand, the two matrices A jAj A jAj
þ A A are similar and so the traces of their nth powers are equal, because A Aþ
I I Aþ A I I A 0 where Id is the identity matrix with the ¼ 0 I A Aþ 0 I A jAj same dimension as A: Thus " trðAn Þ ¼ tr
Aþ A
A Aþ
#n ! trðjAjn Þ:
Therefore, the previous lemma implies the following one. Lemma 11. Let A be an integral square matrix with possibly negative entries, then the sequence ftrðAn Þg is a generalized Fermat sequence. Finally, we are in the position to prove Theorem 5 by applying Lemma 11 to a companion matrix.
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Proof of Theorem 5. Let fan g be the Newton sequence generated by fcn g: Let kX1 and let fbn g be the Newton sequence generated by ci with 1pipk: Then bn ¼ an for all 1pnpk: Let A denote the companion matrix of xk c1 xk1 ? ck1 x ck ; then bn ¼ trðAn Þ for all nAN: By the previous lemma, fbn g is a generalized Fermat sequence. Therefore, for 1pnpk; X X mðdÞan ¼ mðdÞbn is divisible by n: djn
d
djn
d
Since kX1 is arbitrary, we have that fan g is a generalized Fermat sequence.
&
Acknowledgments The authors thank Professor Peter Jau-Shyong Shiue for invaluable suggestions which led to improvements in this paper.
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http://www.elsevier.com/locate/jnt
Generalized Fermat, double Fermat and Newton sequences Bau-Sen Du,a Sen-Shan Huang,b and Ming-Chia Lib b
a Academia Sinica, Institute of Mathematics, Taipei 115, Taiwan 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; Mo¨bius 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 double Fermat sequences. Definition 1. Let fan gN n¼1 be a sequence of integers and be simply denoted by fan g: We call fan g 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
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(resp. for every prime number n), X djn
173
mðdÞan 0 ðmod nÞ; d
where m is the Mo¨bius function, that is, mð1Þ ¼ 1; mðmÞ ¼ ð1Þk if m is a product of k distinct prime numbers, and mðmÞ ¼ 0 otherwise. We call fan g a double Fermat sequence if the following conditions are satisfied: P 1. djn; d odd mðdÞan 0 ðmod 2nÞ for any nAN with an odd prime factor. d
2. a2k 1 ðmod 2kþ1 Þ for any kAN,f0g:
Fermat’s little theorem states that fan g with a an integer is a Fermat sequence. Many generalized Fermat sequences can be obtained by counting numbers of periodic points for maps (see [7–11,16,17] for interval maps and see Section 5 for edge-shift maps). Double Fermat sequences arise naturally from the numbers of symmetric periodic points for odd interval maps (refer to [6,8,10]), although the definition seems artificial at first sight. In Section 2, we give equivalent criteria for generalized Fermat and double Fermat sequences (Theorem 3). As applications of it, we can show that (i) double Fermat sequences must be generalized Fermat sequences; (ii) double Fermat sequences can be characterized as generalized Fermat sequences equipped with some congruence property; (iii) every generalized Fermat sequence and every double Fermat sequence consist of infinitely many generalized Fermat subsequences and double Fermat subsequences respectively; and (iv) the term-by-term product of two generalized (resp. double) Fermat sequences is also a generalized (resp. double) Fermat sequence. Next, we give the definition of Newton sequences generated by sequences of integers, which naturally extends the definition of the usual Newton sequences generated by finitely many integers as in [13]. Definition 2. Given a sequence of integers fcn g; the Newton sequence fan g generated by fcn g is defined by the Newton identities, namely, an ¼ c1 an1 þ c2 an2 þ ? þ cn1 a1 þ ncn : If there exists kAN such that cn ¼ 0 for all n4k; then we simply call fan g the Newton sequence generated by finitely many integers ci with 1pipk: It is known that the Newton sequence fan g generated by ci with 1pipk satisfies an ¼ trðAn Þ; where A is the companion matrix of the polynomial xk c1 xk1 c2 xk2 ? ck1 x ck and trðAn Þ is the trace of An (refer to [12]). The Newton sequence generated by the sequence itself is of the form fð1Þn1 an g; this is also a simple example of a Newton sequence not generated by finitely many integers.
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In Section 3, we prove that every Newton sequence generated by integers is a generalized Fermat sequence, and vice versa (Theorems 5 and 6). As an application to congruence identities in number theory, we show that many well-known sequences, e.g., the Lucas sequences and the Lucas functions, are generalized Fermat sequences. In Section 4, we show that every double Fermat sequence is a Newton sequence generated by a sequence of integers with the first term odd and all the other terms even, and vice versa (Theorem 9). An application to Waring’s formula in algebraic combinatorics is also shown. In Section 5, we give the proof of Theorem 5 by using symbolic dynamics rather than other possible ways, e.g., p-adic analysis. The reason we use the approach of symbolic dynamics is to further investigate the intimate relation between numbers of periodic 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 concise introduction of symbolic dynamics is included. 2. Generalized Fermat and double Fermat sequences In this section, we study properties of generalized Fermat and double Fermat sequences. First, we have criteria for generalized Fermat sequences and double Fermat sequences. Theorem 3. Let fan g be a sequence of integers. Then 1. fan g 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 an ðmod pt Þ: p
2. fan g 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 an ðmod 2sþ1 pt Þ p
a2k 1 ðmod 2kþ1 Þ:
and
Proof. We give the proof of item 1 and omit a similar proof of item 2. Let nAN and p be a prime factor of n; then X X X mðdÞan ¼ mðdÞan þ mðdÞan djn
d
djn; p[d
¼
X
djn; p[d
¼
X
djn; p[d
d
djn; pjd
mðdÞan þ d
X
djn; p[d
mðdÞðan a n Þ d
dp
d
mðdpÞa n
dp
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and so X djn
X
mðdÞan ¼ an an þ d
p
djn; p[d; da1
mðdÞðan a n Þ: d
dp
If pt jj n then one has pt jj dn 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 Fermat sequences. Corollary 4. The following statements hold: 1. Every double Fermat sequence is a generalized Fermat sequence. 2. Let fan g be a generalized Fermat sequence. Then fan g is a double Fermat sequence if and only if a2s m 1 ðmod 2sþ1 Þ for any sAN,f0g and any odd mAN: 3. Let fan g be a generalized (resp. double) Fermat sequence, then for any kAN the sequence fank g 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. We prove 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 an ðmod 2sþ1 Þ p
for any nAN; any odd prime factor p of n and any sX0 with 2s jj n; because of items 1 and 2 of the previous theorem. By applying the hypothesis to m ¼ 2ns and 2ns p resp., one gets that an 1 ðmod 2sþ1 Þ and an 1 ðmod 2sþ1 Þ resp., which yield the desired p
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 Fermat sequences and Newton sequences are essentially the same. Here we state the first one but postpone its proof to Section 5. Theorem 5. Every Newton sequence generated by integers is a generalized Fermat sequence.
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In [4], Dickson showed that every Newton sequence generated by finitely many integers with zero as the first generator is a Fermat sequence. In [12], Gillespie extended the result by waving the restriction on the first generator. Theorem 5 is a generalization 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 Newton sequences generated by integers and so, by Theorem 5, they are generalized Fermat sequences. Some of these results are new. 1. The sequence fan g 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 is the celebrated Fermat’s little theorem. 2. The k-Lucas sequence fan g; defined by an ¼ 1 for 1pnpk 1; ak ¼ k þ 1; and an ¼ an1 þ ank for nXk þ 1; is the Newton sequence generated by c1 ¼ 1; c2 ¼ c3 ¼ ? ¼ ck1 ¼ 0; ck ¼ 1: In particular, the 2-Lucas sequence 1; 3; 4; 7; 11; y is a generalized Fermat sequence, as given in [21]. 3. A special type of the generalized k-Fibonacci sequence fan g; defined by an ¼ 2n 1 for 1pnpk and an ¼ an1 þ an2 þ ? þ ank for nXk þ 1; is the Newton sequence 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 ¼ 2an1 anðkþ1Þ for all nXk þ 2; and so the sequence fan g 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 Ln ðy; zÞ with nX1 are defined by L1 ðy; zÞ ¼ y; L2 ðy; zÞ ¼ y2 2z; and Ln ðy; zÞ ¼ yLn1 ðy; zÞ zLn2 ðy; zÞ for nX3; (refer to Dickson’s elaborate book [5, Chapter XVII]). For integers y and z; the sequence fLn ðy; zÞg is the Newton sequence generated by c1 ¼ y and c2 ¼ z and so is a generalized Fermat sequence. In particular, when y is even, say y ¼ 2m; and z ¼ 1; the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi functions Pn ðmÞ ¼ Ln ð2m; 1Þ ¼ ðm þ m2 þ 1Þn þ ðm m2 þ 1Þn are the socalled Pell–Lucas polynomials. Another special case when y ¼ 2m is even and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ 1; the functions Tn ðmÞ ¼ 12Ln ð2m; 1Þ ¼ 12ðm þ m2 1Þn þ 12ðm m2 1Þn are called the Tchebycheff polynomials of the first kind and so the sequence f2Tn ðmÞg 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 fan g be a generalized Fermat sequence. Define the sequence fcn g by c1 ¼ a1 and the recurrence relation cn ¼
an ðc1 an1 þ c2 an2 þ ? þ cn1 a1 Þ : n
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177
It is evident that fan g is the Newton sequence generated by fcn g: We show that all the cn are integers by induction on n: For n ¼ 1; c1 ¼ a1 is an integer. Assume that each of c1 ; c2 ; y; ck is an integer. Let fbn g be the Newton sequence generated by ci with 1pipk: Then we get recurrently that for 1pmpk; bm ¼ c1 am1 þ c2 am2 þ ? þ cm1 a1 þ mcm ¼ am ; and so bkþ1 ¼ c1 ak þ c2 ak1 þ ? þ ck a1 : The assumption here and Theorem 5 imply that both fan g and fbn g are generalized Fermat sequences. Therefore, k þ 1 divides both X
akþ1 þ
mðdÞakþ1
djðkþ1Þ; da1
d
and
bkþ1 þ
X
mðdÞbkþ1 :
djðkþ1Þ; da1
d
Since am ¼ bm for all 1pmpk; the last two sums are identical and so k þ 1 divides bkþ1 is an integer. The induction has been akþ1 bkþ1 : Therefore, ckþ1 ¼ akþ1kþ1 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 Newton sequences. First, we give an explicit formula for each term of a Newton sequence in terms of its generators (cf. [2] and [22]). Proposition 7. Let fan g be the Newton sequence generated by fcn g; then for any nAN;
an ¼
X k1 þ2k2 þ?þnkn ¼n
! k1 þ k2 þ ? þ kn k1 k2 n c1 c2 ?cknn ; k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
2 þ?þkn Þ! 2 þ?þkn Þ ¼ ðk1kþk with 0! ¼ 1 by convention and the symbol 00 ; if it where ðk1kþk 1 ;k2 ;y;kn 1 !k2 !?kn ! occurs, is interpreted as the value 1.
Proof. Fix nAN and let A denote the companion matrix of xn c1 xn1 ? cn1 x cn ; then an ¼ trðAn Þ: By Liouville’s formula (refer to [20]), we have that N X trðAm Þ m 1 t ¼ log m detðI tAÞ m¼1 1 ¼ log 1 c1 t c2 t 2 ? cn t n N X 1 ðc1 t þ c2 t2 þ ? þ cn tn Þk ¼ k k¼1
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! ) k1 þ k2 þ ? þ kn ck11 ck22 ?cknn ¼ tk1 þ2k2 þ?þnkn k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn k¼1 k1 þk2 þ?þkn ¼k ( ! ) N X X k1 þ k2 þ ? þ kn ck11 ck22 ?cknn ¼ tm : k þ k þ ? þ k k ; k ; ?; k 1 2 n 1 2 n m¼1 k þ2k þ?þnk ¼m N X
(
X
1
n
2
By equating the coefficient of tn ; we obtain that ! k1 þ k2 þ ? þ kn k1 k2 n an ¼ trðA Þ ¼ c1 c2 ?cknn : k þ k þ ? þ k k ; k ; ?; k 2 n 1 2 n k1 þ2k2 þ?þnkn ¼n 1 X
n
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 above proposition is relevant to the so-called Waring’s formula in the theory of algebraic combinatorics [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 ¼ P Pn i1 pi eni and nhn ¼ ni¼1 pi hni (refer to [1,14]). By the above proposition, i¼1 ð1Þ we get that the following two expressions are both equal to pn in terms of ei ’s and hi ’s, respectively: X
ð1Þ
k2 þk4 þ?þk n 2½ 2
k1 þ2k2 þ?þnkn ¼n
k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
!
n k1 þ k2 þ ? þ kn
ek11 ek22 ?eknn
and X k1 þ2k2 þ?þnkn ¼n
ð1Þk1 þk2 þ?þkn1
n k1 þ k2 þ ? þ kn
! k1 þ k2 þ ? þ kn k1 k2 h1 h2 ?hknn : k1 ; k2 ; ?; kn
Additionally, in the third example below Theorem 5, we have that ci ¼ 1 with 1pipn generates the Newton sequence having the nth term 2n 1; and so the above
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proposition implies that X k1 þ2k2 þ?þnkn ¼n
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 Newton sequence with an additional restriction on the generators. Theorem 9. Every double Fermat sequence is a Newton sequence generated by fcn g with 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 the Newton sequence fan g generated by fcn g; the following two statements are equivalent: (S1) a2k m 1 ðmod 2kþ1 Þ 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 2k m41 with m odd. By the previous proposition, we have
k
X
2k m k þ k2 þ ? þ kn k1 þ2k2 þ?þnkn ¼n; k1 on 1 ! k1 þ k2 þ ? þ kn k1 k2 c1 c2 ?cknn : k1 ; k2 ; y; kn
an ¼ cn1 þ
It is easy to see that c21 m 1 ðmod 2kþ1 Þ 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 k m 2 þ?þkn such that 2b divides k1 þk22þ?þk ðk1kþk Þck11 ck22 ?cknn : Then, it remains to show that n 1 ;k2 ;?;kn bXk þ 1 all the time. By the assumption on parity of cn and de Polignac’s formula from elementary number theory [19], we have ! X N
N X k1 þ k2 þ ? þ kn 1 k1 bX k þ 2j 2j j¼1 j¼1 ! ! ! N N N X X X k2 k3 kn þ k2 þ k3 þ ? þ kn 2j 2j 2j j¼1 j¼1 j¼1 ! ! ! N N N X X X k2 k3 kn p k þ k2 þ k þ ? þ k 4k; 3 n j j 2 2 2j j¼1 j¼1 j¼1
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where ½x denotes the greatest integer less than or equal to x and the last inequality holds since k1 on and then there exists some ki with i41 such that ki 40: 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 2k m41 with kX0 and odd mX1 such that cn is odd. By the previous proposition, we obtain that X
ncn ¼ an cn1
k1 þ2k2 þ?þnkn ¼n; k1 on;
k1 þ k2 þ ? þ kn k1 ; k2 ; y; kn
!
n k þ k2 þ ? þ kn k ¼0 1 n
ck11 ck22 ?cknn :
In the last equality, the term ncn ð¼ 2k mcn Þ is divisible by 2k ; but not divisible by 2kþ1 : Moreover, an cn1 0 ðmod 2kþ1 Þ since an 1 ðmod 2kþ1 Þ 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 prove Theorem 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 with nonnegative integer entries. The graph of A is the graph GA with state set S ¼ f1; 2; y; kg and with AIJ distinct edges from edge set E with initial state I and terminal state J: The edge shift space SA is the space of sequences of edges from E specified by SA ¼ fe0 e1 e2 ?jei AE and tðei Þ ¼ iðeiþ1 Þ for all integers iX0g: The shift map sA : SA -SA is defined to be sA ðe0 e1 e2 e3 ?Þ ¼ e1 e2 e3 ?: For nAN; let snA denote the composition of sA with itself n times. A point e ¼ % e0 e1 e2 ?ASA is called a period-n point for sA if snA ðeÞ ¼ e and sjA ðeÞae for 1pjpn % ðs Þ denote 1: Let Pern ðsA Þ denote the set of all period-n points% for %sA and let% #Per n A the cardinal number of the set Pern ðsA Þ: It is clear that #Pern ðsA Þ is finite and divisible by n; and moreover, #Per1 ðsnA Þ ¼ P djn #Perd ðsA Þ: According to the Mo¨bius inversion formula or the inclusion–
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n P exclusion principle (refer to [19]), we have #Pern ðsA Þ ¼ djn mðdÞ#Per1 ðsdA Þ: Thus, the sequence f#Per1 ðsnA Þg is a generalized Fermat sequence. On the other hand, the fact that trðAn Þ ¼ #Per1 ðsnA Þ can be easily proved as follows. Let S be the state set and let E denote the edge set from the graph of A: A finite sequence of edges from E; denoted by p ¼ e0 e1 ?em ; is called a path of length m þ 1 from iðe0 Þ to tðem Þ if tðei Þ ¼ iðeiþ1 Þ for 0pipm 1: For nAN; let ðAn ÞIJ be the ðI; JÞth entry of An and let Pðn; I; JÞ be the number of paths of length n þ 1 from I to J: By induction on n; one can show that for every nAN; ðAn ÞIJ ¼ Pðn; I; JÞ for all choices of states I and J in S: Hence,
trðAn Þ ¼
X
ðAn ÞII ¼
IAS
X
Pðn; I; IÞ ¼ #Per1 ðsnA Þ:
IAS
So, we have shown the following lemma. Lemma 10. Let A be a square matrix with nonnegative integer entries, then the sequence ftrðAn Þg is a generalized Fermat sequence. Next, we consider integral matrices with possibly negative entries. For a matrix A 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 trðAn Þ þ n
A 0 A 0 n trðjAj Þ ¼ tr and : On the other hand, the two matrices A jAj A jAj
þ A A are similar and so the traces of their nth powers are equal, because A Aþ
I I Aþ A I I A 0 where Id is the identity matrix with the ¼ 0 I A Aþ 0 I A jAj same dimension as A: Thus " trðAn Þ ¼ tr
Aþ A
A Aþ
#n ! trðjAjn Þ:
Therefore, the previous lemma implies the following one. Lemma 11. Let A be an integral square matrix with possibly negative entries, then the sequence ftrðAn Þg is a generalized Fermat sequence. Finally, we are in the position to prove Theorem 5 by applying Lemma 11 to a companion matrix.
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Proof of Theorem 5. Let fan g be the Newton sequence generated by fcn g: Let kX1 and let fbn g be the Newton sequence generated by ci with 1pipk: Then bn ¼ an for all 1pnpk: Let A denote the companion matrix of xk c1 xk1 ? ck1 x ck ; then bn ¼ trðAn Þ for all nAN: By the previous lemma, fbn g is a generalized Fermat sequence. Therefore, for 1pnpk; X X mðdÞan ¼ mðdÞbn is divisible by n: djn
d
djn
d
Since kX1 is arbitrary, we have that fan g is a generalized Fermat sequence.
&
Acknowledgments The authors thank Professor Peter Jau-Shyong Shiue for invaluable suggestions which led to improvements in this paper.
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