generalized fermat, double fermat and newton sequences
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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: mabsdu@sinica.edu.tw (B.-S. Du), shuang@math.ncue.edu.tw (S.-S. Huang),
mcli@math.ncue.edu.tw (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.
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183 175
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
:
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183176
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
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183 177
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
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183178
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;
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183 179
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–
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183180
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.
B.-S. Du et al. / Journal of Number Theory 98 (2003) 172–183 181
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.
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