In this paper the m-powers with  m ≥ 2 included in the Lucas sequences when  the index satisfies some conditions are foun

López González, M. Dolores

Rodrigo Hitos, Javier

English

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 1
Powers in the Lucas sequence when the index is divisible by three 
 
Mª Dolores Lópeza,, Javier Rodrigob 
 
aDepartamento de Matemática e Informática Aplicadas a la Ingeniería Civil de la E.T.S.I. 
Caminos, Canales y Puertos, Universidad Politécnica de Madrid marilo.lopez@ upm.es 
bDepartamento de Matemática Aplicada, E.T.S. de Ingeniería, Universidad Pontificia Comillas 
de Madrid 
 
*Corresponding author: Mª Dolores López, Departamento de Matemática e Informática 
Aplicadas a la Ingeniería Civil de la E.T.S.I. Caminos, Canales y Puertos, Universidad 
Politécnica de Madrid, c/ Profesor Aranguren s/n, Madrid 28040, Spain 
Tf: 34- 913366669; Fax: 34-913366782 
 2
Abstract: 
In this paper the m-powers with 2≥m  included in the Lucas sequences when 
the index satisfies some conditions are found. 
 
Keywords: Number theory, Lucas sequences, Diophantine equations. 
 
 3
1. Introduction 
The Lucas and companion Lucas sequences have been widely studied in the 
scientific literature as a natural extension of the so called Fibonacci sequence. They are 
defined inductively by: 0 ,0 =QPU , 1 ,1 =QPU , QPnQPnQPn UQUPU  ,2 ,1 ,   −− −=  and 2 ,0 =QPV , 
PV QP = ,1 , QPnQPnQPn VQVPV  ,2 ,1 ,   −− −=  respectively, where QP  ,  are integer numbers (the 
Fibonacci sequence is the particular case of QPnU
 ,  with 1=P , 1−=Q . The indices  
QP  ,  will be omitted when they are not necessary). 
Concretely, the search of the powers included in these sequences has deserved a special 
attention by the number theorist. The main results at this respect for nU  are: 
- For odd relative primes QP  , , the only square in nU  with 2>n , 6≠n  is 1441 ,112 =−U  
(see Ribenboim, Bremner&Tzanakis) 
- For P  divisible by 4, there is no squares in 1 , −PnU  for positive, even numbers n  when 
additional conditions on P  are considered (Kagawa) 
- The only powers in 1 ,1 −nU  with 2>n  are 81 ,16 =−U  and 1441 ,112 =−U  (Bugeaud) 
- The only power in 1 ,2 −nU  with 1>n  is 1691 ,27 =−U  (Cohn) 
In this paper the case 0>P , 1−=Q  is studied: it is proved that, for n  divisible by 
three and satisfying some extra conditions, the powers in nU  are always attained for 
1=P  and they are 81 ,16 =−U  and 1441 ,112 =−U . To do this, the diophantine equation 
mP
n uU =−1 ,  with unknowns ( )muPn ,,, , 2≥m , is related with these diophantine 
equations according to the different cases: 
- The Catalan equation 12 =− yxm  
- The Thue equation 12 =− mm yx  
- The equation 12 42 =− yx  
The following notation applies in the paper:  
- N  for the set of natural numbers 
- ( )baGCD ,  for the greatest common divisor of the integer numbers ba,  
- ba  if the integer number a  divides to the integer number b  
- ( )mba mod≡  if a  is congruent with b  modulus m  (that is to say, abm − ) 
 4
- ⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
n
 for the binomial coefficient. 
- ( )nep  for the exponent of the prime p  in the factorization in prime numbers of the 
integer number n  (if p  is not a prime factor of n , it is assumed that ( ) 0=nep ). 
The following two identities for nU , nV , kU , kV  are used along the paper (see 
Kagawa): 
( ) ( )knkknnkn PUVPUV 4 4 2 221 ++=++−  when nk  and k , n  are positive numbers  
(1) 
( ) ( )( ) ( )kkk UPV 1 4 4 222 −=+−         (2) 
 
The paper is structured as follows: in section 2 the main result is stated and in section 3 
some concluding remarks are given. Along section 2, k  and n  are positive integer 
numbers. 
 
2. Development 
To prove the result mentioned in the introduction, a lemma is needed: 
Lemma 
If nk , then 
k
nU
U
UGCD k
k
n  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
 
Proof 
First it is noted that if nk , then nk UU  (see for example Ribenboim), so 
k
n
U
U
 is an 
integer number and the statement makes sense. Now, the following cases appear: 
1) If kV  is an odd number, then by expanding (1) it is obtained that: 
( ) ( )⎟⎟⎠
⎞
⎜⎜⎝
⎛ +=++⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+= −−−− kk
n
kkk
k
n
kk
k
n
kn
k
n
UfV
k
nUPUVk
n
UV
k
nU
123311   ...4   
3
  2 , where 
 5
( )xf  is a polynomial without independent term. Now, since kk
k
n UU
U
UGCD  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
, 
( )
1
1
2
 ,
−
− +
=⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
n
k
k
n
k
k
n
k
k
n
UfV
k
n
U
UU
U
UGCD , then ⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
k
n U
U
U
GCD ,  divides to kU ,  
( )kk
n
k UfVk
n +−1 , so it divides to ( ) ( ) 11 −− =−+ knkkkk
n
k Vk
nUfUfV
k
n .  
Since kV  is an odd number, it is satisfied that ( ) 1, =kk UVGCD  by identity (2) and then 
k
nU
U
UGCD k
k
n  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
 as desired. 
2) If kV  and P are even numbers, then (1) and (2) yield respectively: 
( )⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+⎟⎠
⎞⎜⎝
⎛=++⎟⎠
⎞⎜⎝
⎛
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+⎟⎠
⎞⎜⎝
⎛=
−−−
k
k
n
k
kk
k
n
k
k
k
n
k
n Ug
V
k
nUPUVk
n
UV
k
nU
12
3
31
2
  ...
4
4  
2
 
3
 
2
  
( )kkk UPV 14
4
2
2
22
−=+−⎟⎠
⎞⎜⎝
⎛  
Where ( )xg  is a polynomial with integer coefficients and without independent term. 
The last identity implies that 1,
2
=⎟⎠
⎞⎜⎝
⎛
k
k UVGCD . 
Now, since kk
k
n UU
U
UGCD  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
, ( )kk
n
k
k
n
k
k
n UgV
k
n
U
UU
U
UGCD +⎟⎠
⎞⎜⎝
⎛=⎟⎟⎠
⎞
⎜⎜⎝
⎛ −1
2
 , , then 
⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
k
n U
U
U
GCD ,  divides to ( ) ( ) 11
22
−−
⎟⎠
⎞⎜⎝
⎛=−+⎟⎠
⎞⎜⎝
⎛ k
n
k
kk
k
n
k V
k
nUgUgV
k
n , so 
k
nU
U
UGCD k
k
n  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
 also in this case because 1,
2
=⎟⎠
⎞⎜⎝
⎛
k
k UVGCD .  
3) If kV  is an even number and P is an odd number, then (2) implies that kU  is an even 
number, so (2) can be rewritten as ( ) ( )kkk UPV 1
2
 4
2
2
2
2
−=⎟⎠
⎞⎜⎝
⎛+−⎟⎠
⎞⎜⎝
⎛  and 
2
kV ,  
2
kU  have 
different parity. Identity (1) can be expressed in this case as: 
 6
k
n
kknn PUVPUV ⎟⎠
⎞⎜⎝
⎛ ++=++ 4 
22
4 
22
22 .       (3)  
In an analogous way to case 1), it can be seen that 
⎟⎟
⎟⎟
⎠
⎞
⎜⎜
⎜⎜
⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛ k
U
U
e
k
n
UU
U
GCD
k
n
,
2
2
 divides to 
k
n , so 
in order to prove that 
k
nU
U
UGCD k
k
n  , ⎟⎟⎠
⎞
⎜⎜⎝
⎛
, it is sufficient to see that 
⎟⎠
⎞⎜⎝
⎛≤⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
neU
U
UGCDe k
k
n
22 , . Three new cases appear: 
- If 
k
n  is an odd number, then expanding the right side of (3) and equaling the terms 
with 42 +P , it is obtained that: 
( ) ( ) 21 1212231 4 
2
...4 
2
 
2
 
32
 ⎟⎠
⎞⎜⎝
⎛ −
−−−
+⎟⎠
⎞⎜⎝
⎛+++⎟⎠
⎞⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+⎟⎠
⎞⎜⎝
⎛= k
nk
n
kkk
n
kk
n
k
k
n PUPUVk
n
V
k
n
U
U
, so 
k
n
U
U
 is 
an odd number regardless 
2
kU  is an odd number or 
2
kV  is an odd number and then 
⎟⎠
⎞⎜⎝
⎛==⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
neU
U
UGCDe k
k
n
22 0, . 
-If 
k
n  is an even number and 
2
kV  is an odd number, then 
2
kU  is an even number and it 
is obtained in an analogous way than in the previous case that: 
( ) ( ) 21 2222231 4 
2
 ...4 
2
 
2
 
32
 ⎟⎠
⎞⎜⎝
⎛ −
−−−
+⎟⎠
⎞⎜⎝
⎛+++⎟⎠
⎞⎜⎝
⎛⎟⎠
⎞⎜⎝
⎛
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
+⎟⎠
⎞⎜⎝
⎛= k
nk
n
kkk
n
kk
n
k
k
n PU
k
nPUVk
nV
k
n
U
U
 
 7
Now, it is a known fact that ⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎠
⎞⎜⎝
⎛ i
k
n
i
k
nGCD
k
n
 ,
, so ⎟⎠
⎞⎜⎝
⎛≥
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
⎟⎟
⎟
⎠
⎞
⎜⎜
⎜
⎝
⎛
k
ne
i
k
n
e 22  when i  is an odd 
number. Since 
2
kU  is an even number, this implies that ⎟⎠
⎞⎜⎝
⎛=⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
ne
U
U
e
k
n
22 , so 
⎟⎠
⎞⎜⎝
⎛≤⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
neU
U
UGCDe k
k
n
22 ,  as desired. 
-If 
k
n  and 
2
kV  are even numbers, then 
2
kU  is an odd number, so 
⎟⎠
⎞⎜⎝
⎛≤≤⎟⎟⎠
⎞
⎜⎜⎝
⎛
⎟⎟⎠
⎞
⎜⎜⎝
⎛
k
neU
U
UGCDe k
k
n
22 1, , as desired. 
 
Now the theorem is stated: 
Theorem 
If 1
2
, =⎟⎠
⎞⎜⎝
⎛ nPGCD  when n  is an even number, n  has not prime factors congruent with 
1 modulus 4 and it has the factor three, then the only powers included in nU  are 
81 ,16 =−U  and 1441 ,112 =−U   
Proof 
It is satisfied that ( )
3
2
3
3  1 U
U
P
U
U
UU nnn +== , with 3 1,
2
3
nP
U
UGCD n ⎟⎟⎠
⎞
⎜⎜⎝
⎛ +  by the lemma. 
Hence ⎟⎟⎠
⎞
⎜⎜⎝
⎛ +1, 2
3
P
U
U
GCD n  has not prime factors congruent with 1 modulus 4 since n  
has not prime factors congruent with 1 modulus 4 by hypothesis. It is also satisfied that 
⎟⎟⎠
⎞
⎜⎜⎝
⎛ +1, 2
3
P
U
U
GCD n  has not prime factors congruent with 3 modulus 4 (otherwise, 1−  
would be a quadratic residue modulus a prime number that is congruent with 3 modulus 
4). Therefore, 2or  11, 2
3
=⎟⎟⎠
⎞
⎜⎜⎝
⎛ +P
U
U
GCD n  (since ( )4mod 212 ≡+P  when 12 +P  is an 
even number). So, if  mn uU =  with Nmu ∈ , , 2≥m , then the following cases must be 
considered: 
 8
1) If 11, 2
3
=⎟⎟⎠
⎞
⎜⎜⎝
⎛ +P
U
U
GCD n , then ( ) mnn uUUPU =+= 32  1  implies that mvP =+12  with 
Nv∈ , so the Catalan equation 12 =− yxm  has a solution in positive integers: ( )Pv, , a 
contradiction (see Mihailescu)  
2) If 21, 2
3
=⎟⎟⎠
⎞
⎜⎜⎝
⎛ +P
U
U
GCD n , then n  must be an even number because 
3
21, 2
3
nP
U
UGCD n =⎟⎟⎠
⎞
⎜⎜⎝
⎛ +  by the lemma, so ( )
32
2
32
32  1 UU
U
PP
UU
U
UUU nnn +== . 
Now, ( ) mnn uUUUPPU =+= 322  1  implies that mwP =  with Nw∈ , mvP 212 =+  
with Nv∈ ,  because 21, 2
3
=⎟⎟⎠
⎞
⎜⎜⎝
⎛ +P
U
U
GCD n  and 1,
2
=⎟⎟⎠
⎞
⎜⎜⎝
⎛
U
U
PGCD n  by the lemma and 
the hypothesis 1
2
, =⎟⎠
⎞⎜⎝
⎛ nPGCD . Then it is satisfied that ( ) 12 2 =− mm wv , so the 
diophantine equation 12 =− mm yx  has a solution in positive integers: ( )2, wv . But for 
3≥m  the only solution in positive integers to this equation is 1== yx  (see Darmon). 
This implies that 12 =w  and then 1== mwP  (P  is a positive number). But in this case 
the only m -power with 3≥m  in nU  for the required conditions on n  is 31 ,16 28 ==−U , 
as stated in the introduction. 
For 2=m  it is satisfied that 12 42 =− wv , so the diophantine equation 12 42 =− yx  
has a solution in positive integers: ( )wv, . But the only solution in positive integers to 
this equation is 1== yx  (see M. Le). This implies that 1== mwP . But in this case the 
only square in nU  for the required conditions on n  is 
21 ,1
12 12144 ==−U , as stated in the 
introduction and the proof is finished. 
 
3. Concluding remarks 
A result about the powers in the Lucas sequences nU  has been stated for indices n  
divisible by three and satisfying certain conditions. This result is very general in the 
sense that it has been stated for m-powers for every 2≥m , and without restrictions on 
the parameter P  when n  is an odd number. Therefore this paper complements previous 
 9
studies of Kagawa, Ribenboim about the squares in the Lucas sequences and of Bennett  
about the m-powers with 3≥m  in the Lucas sequences. 
A future line of work could be to relax the conditions on the index n  in the theorem of 
section 2 
 
References 
1. M. A. Bennett, Powers in recurrence sequences: Pell equations, Transactions of the 
American Mathematical Society, 357 (2004), 1675–1691 
2. A. Bremner and N. Tzanakis, Lucas sequences whose 12th or 9th term is a square, J. 
Number Theory, 107 (2004), 215-227.  
3. Y. Bugeaud, M. Mignotte and S. Siksek, Classical and modular approaches to 
exponential Diophantine equations, I. Fibonacci and Lucas perfect powers. Annals 
of Math. 163 (2006), 969-1018. 
4. Cohn, J. H. E. (1996). Perfect Pell powers. Glasgow Mathematical Journal 38 (1): 
19–20. 
5. H. Darmon and L. Merel, Winding quotients and some variants of Fermat’s Last 
Theorem, J. Reine Angew. Math. 490 (1997), 81–100. 
6. Kagawa, T. and Terai, N.: Squares in Lucas sequences and some Diophantine 
equations. Manuscripta Math. 96, 195–202 (1998). 
7. M. Le. On the diophantine equation ( )( ) 21 1 yxx nm =++ . Acta arithmetica, 
LXXXII.1, 17-26, 1997. 
8. Mihailescu, P. Primary ciclotomic units and a proof of Catalan’s conjecture; J. 
Reine angew.Math. 572, 167-195, 2004. 
9. P. Ribenboim and W. L. McDaniel, The square terms in Lucas sequences. J. 
Number Theory 58 (1996), 104–122. 
 


Powers in the Lucas sequence when the index is divisible by three

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