Periodic Patterns and Block Structures in Squared Pell Sequence Modulo 10e
- Rima Patel
- Devbhadra Shah
- 1165-1175
- Jun 13, 2025
- Mathematics
Periodic Patterns and Block Structures in Squared Pell Sequence Modulo 10e
Rima Patel1, Devbhadra Shah2
1Department of Humanities and Social Science, Mahavir Swami College of Polytechnic, Bhagwan Mahavir University, Surat, India
2Department of Mathematics, Veer Narmad South Gujarat University, Surat, India
DOI: https://doi.org/10.51244/IJRSI.2025.120500113
Received: 20 May 2025; Accepted: 30 May 2025; Published: 13 June 2025
ABSTRACT
In this paper, we investigate the periodic properties of the squared Pell sequence {
}, which is defined by the recurrence relation 
for all 
; with 
, where 
denotes 
th Pell number. For any modulus 
, we introduce a novel concept of ‘blocks’ within this sequence by examining the distribution of residues over a single period of the squared Pell sequence. Our results reveal that the length of any given period of the squared Pell sequence comprises either 1 or 2 blocks.
Keywords: Fibonacci sequence, Pell sequence, Periodicity of Pell sequence.
INTRODUCTION
The Fibonacci sequence 
shows interesting periodic properties under modulo 
. Initially, the last digits of Fibonacci numbers seem random, but a clear pattern emerges: the sequence of last digits repeats every 60 numbers. Therefore, the last digits exhibit a periodicity with a cycle length of 60, expressed as 
for any 
, where 
. Koshy [7] proved this using mathematical induction.
In 
, Kramer and Hoggatt Jr. 
established the periodicity of Fibonacci sequence as well as of Lucas sequence when considered modulo 
. Patel, Shah [5] considered the periodicity of generalized Lucas numbers and proved the result when the length of its period under modulo 
.
This brings in to mind an immediate question – For any given positive integer , does the sequence is periodic when considered modulo 
? In 
, Wall 
examined the periodic nature of with respect to any positive integer and showed that consistently exhibits periodicity.
Ömür Deveci, Erdal Karaduman 
proved some elementary results for the periodicity of 
. For further details about Pell numbers, one can refer Horadam 
and Koshy 
.
This listing can be further extended as several articles are available in the literature concerning the periodicity of varied generalizations of the Fibonacci sequence. In the following section, we now consider the periodicity of a new sequence – the squared Pell sequence.
SQUARED PELL SEQUENCE
The squared Pell sequence is the sequence which consists of the squares of all the Pell numbers in order.
Definition: The sequence 
represents the squares of corresponding terms of the sequence in order. In other words, 
; for all 
, where stands for th Pell number.
It is trivial to note that 
. We first derive some elementary results for this sequence which will be used further in this paper. The following result gives a recurrence relation which helps to reduce the terms of into smaller terms.
Lemma 
:
.
Lemma 
: 
.
Lemma 
: 
.
In the following section we study the periodicity of sequence and obtain some interesting results related with its residues.
PERIODICITY OF SQUARED PELL SEQUENCE
In this section, we study in detail about the periodic nature of when considered modulo . For the detailed insights, one can refer Marc 
Definition: By 
, we mean the sequence of the least non-negative residues of the terms of the squares of terms of the sequence in order taken modulo
.
As an illustration, we consider 𝑆𝑃(𝑚𝑜𝑑 8) in the following table:
Table 𝟒. 1: 𝑺𝑷(𝒎𝒐𝒅 𝟖)
 |
0 | 1 | 2 | 3 | 4 | 5 | 6 |
|
0 | 1 | 4 | 25 | 144 | 841 | 4900 |
 |
0 | 1 | 4 | 1 | 0 | 1 | 4 |
From the above table, it can be noticed that the sequence 𝑆𝑃(𝑚𝑜𝑑 8) is periodic. Furthermore, it is not difficult to check that 𝑆𝑃4𝑛+𝑖 ≡ 𝑃𝑖(𝑚𝑜𝑑 8); where 𝑛 ≥ 0. This clearly indicates that the period of 𝑆𝑃(𝑚𝑜𝑑 8) is 4.
We now prove several results for the periodic nature of analogues to that of 
.
Lemma
: The sequence is always periodic; for any integer and its starting values 
.
We next introduce the notation for the length of period of 
.
Definition: 
denotes the length of period of the squared Pell sequence modulo .
The following are some immediate consequences from the lemmas 
, 
and the definition of .
Lemma 
: (a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
, 
.
Fact 
: Since is periodic, we will often use the fact that ‘if both 
and 
holds, then 
.
Lemma 
: For any given integer , there are infinitely many squared Pell numbers which are divisible by .
Theorem 
: If 
then 
.
Theorem 
: 
, for various values of , where 
and
’s are distinct primes.
Theorem
:
.
VALUE OF 
In this section, we obtain the value of 
when 
.
Theorem 
: 
.
Proof: We notice that 
. Therefore, 
. For 
, we prove the result by induction.
We note that 
and 
. Therefore, 
and 
. This proves the result for 
. We assume that the result holds for some positive integer 
. Thus,
; 
. 
Then by the lemma 
and 
, we have
By lemma 
, we have 
. By taking 
and using , we have
Thus,
Again, by lemma 
, we have 
. Considering , we get
. 
But by , we get 
and 
. Thus, 
and 
. By considering modulo 
, we have 
or 
; and 
or 
. Thus by , we have
Thus,
Then by the 
and fact 
, we have
Since 
implies 
, we get
Then by combining equation and , we get
or 
.
We shall show that the case is not possible. In fact, we will show that 
. More precisely, we will prove that
. 
Considering 
, we have (i) 
and (ii) 
. Therefore, is true for . Let it be true for some integer 
. Thus, 
. Considering modulo , we get
or 
Then, 
or 
.
Now since , we have 
.
Also, 
.
This gives,
. 
We also assume that
(This is because if it is not true then replacing 
by 
, we can say that 
is not true. Thus, 
which we need to prove.) Taking modulo , we get 
or . Thus
Now, by lemma , we have 
. Considering 
, we get
By 
and , we thus have
Thus, 
This now confirms that ; that means is not possible. Hence 
. This proves the theorem by induction.
VALUE OF 
In this section, we obtain the value of for the case 
.
Theorem 
: 
Proof: To prove the required result, it is sufficient to prove that
We use induction to prove these results. For 
, we have
and 
.
Thus, 
is true for . We next assume that results hold for some positive integer 
. That is, let the following holds:
We prove that holds for 
also. Therefore, we need to prove that 
and 
. Now, by lemma 
, we have 
. By considering 
, we get
.
Also, by Koshy 
, we have 
and by lemma 
, we have 
. Therefore, 
. Thus,
By Koshy , we have 
.
By considering , we get 
. Now, by lemma 
, we have 
. Also, by lemma 
, we have . Therefore,
Thus,
Using the 
and fact , we can now conclude that
Since 
implies that 
. Also, by induction hypothesis, we have . This gives
Thus using and 
, we conclude that
or 
.
We finally confirm that the case is not possible. In fact, we show that 
. Now by Koshy , we get
.
Considering 
, we get
.
Since by Koshy , we have 
and 
and thus in modulo 
, we have 
and 
. Also, by lemma and 
, we have 
and 
.
Thus, in modulo , we have 
and 
. We get
Therefore, 
. This shows that 
is not possible. Hence, 
. Thus, 
is true for every positive integer 
, which proves the required result.
Finally, using theorem 
and 
we easily conclude the following important result.
Theorem 
: 
.
The following result calculates the period of when considered modulo .
Theorem 
: 
and 
; where 
and 
is any integer.
In the next section, we introduce the notion of blocks within the period of the squared Pell sequence.
BLOCKS WITHIN THE PERIOD OF SQUARED SEQUENCE
In this final section, we study the nature of the blocks within the residues of the squared Pell sequence when considered modulo . We also discuss the distribution of residues within a single period of . For the detailed insights, one can refer Patel, Shah [4].
Definition: 
denotes the smallest positive value of index of squared Pell numbers such that and 
; when 
.
Thus, 
. We call to be the restricted period of . Thus indicates the position of ending of first block which occurs in . We call the finite sequence 
to be the first block occurring in .
Definition: When 
, we call to be without restricted period.
To illustrate these definitions, we consider the following examples.
(i) Since 
, then clearly 
. In this case 
will be without restricted period.
(ii) Since 
, we have 
and 
. Thus, 
. Here 
is the first block in 
.
(iii) Since 
, thus we have 
and 
. Thus, 
. Here 
is the first block in 
.
From last two illustrations, it is seen that the subscript of terms for which and contains equal number of (that is number of) terms and the subscripts are in arithmetic progression with common difference . That is, 
and 
.
Thus, we can say that 
, for each positive integer 
. Moreover, since 
, we say that 
where is some positive integer. Thus, 
.
To illustrate this, we consider
.
Then it can be seen that 
, 
and 
. Thus, in this case and 
.
Later we will show that the value of is always either 
or 
. The following result gives interesting outlook about the divisibility property of suffix .
Lemma 
: 
if and only if 
.
Proof: Let . Then, we have 
; for some 
.
In view of the above comment, 
. This gives .
To prove the converse part, assume that . Then by the definition of , either 
or 
. If then is true and if then as lies in the simple arithmetic progression with first term 
and common difference , we have 
. Therefore, is true in any case. This completes the proof.
The following interesting divisibility property always holds for any arbitrary values of and .
Theorem 
: 
.
Proof: By the definition of , we have 
. Therefore, 
is always true. Thus, 
also holds. Now for any multiple of , 
th position within the list of residues for will always contain zero. Thus, 
; that is 
. Hence, 
, as required.
To illustrate this, we consider 
, In this case we observe that 
. When we consider
,
we observe that 
. Thus, 
.
Definition: By 
, we mean the first positive residue appearing after the blocks in . That is 
and is the smallest such number.
Since, 
and , we have 
. Thus, acts like a multiplier of the first periodic part of .
To illustrate this, we consider 
. Then since 
, we have
Thus, 
.
Definition: 
denote the order of 
.
That is 
and if 
then 
.
As an illustration, if we once again consider , then and 
. Thus, 
.
To illustrate above definitions, we consider the following two examples:
- Since
, clearly 
. Also, the restricted period 
and multiplier 
. Thus, the order of 
and hence 
. - Since
, then clearly , and 
. Since 
, we get 
.
The following resembles the theorem 
for the sequence .
Theorem 
: 
Proof: Throughout the proof we consider all the congruences modulo . Suppose that one period of is partitioned into smaller and finite subsequences 
as shown below:
where 
and every 
contains exactly one .
Clearly each subsequence 
has 
terms and 
. Also, in any , there is exactly one zero. Hence every subsequence is a multiple of 
. More precisely, we have the following congruences:
Now the last term in 
is 
and that of is 
. Also, we have 
. Therefore, 
. By the similar arguments, we have
Therefore, we have
⋮
.
Therefore, 
.
Since the order of is 
, we rewrite sequence 
as follows:
;
with 
.
Thus can be interpreted as the number of blocks in a single period of . It now follows easily that 
.
The following results will be helpful for the study of blocks within the residues of .
Corollary 
:
.
Proof: From above theorem, we have 
and 
. Thus, we have
This shows that the 
term of 
is equal to 
times the 
term of , when considered modulo . Also, from the definition of 
we conclude that 
, when considered modulo . Therefore, from 
and above arguments, we can say that 
. This finally gives
.
Corollary 
: 
.
Proof: Since 
, we write 
, as required
Theorem 
: 
or , for 
.
Proof: By Koshy , we have 
. Taking 
, we get
. 
Now, . Also, we know that
and
.
Therefore, by 
, we have 
. Thus, we get 
, that is 
. Since order of is , we finally conclude that must divide . Hence, for any , we have 
or .
We conclude by presenting a table displaying the values of 
and for 2 ≤ m ≤ 20.
 |
 |
 |
 |
| 2 | 2 | 2 | 1 |
| 3 | 4 | 4 | 1 |
| 4 | 2 | 2 | 1 |
| 5 | 6 | 3 | 2 |
| 6 | 4 | 4 | 1 |
| 7 | 6 | 6 | 1 |
| 8 | 4 | 4 | 1 |
| 9 | 12 | 12 | 1 |
| 10 | 6 | 6 | 1 |
| 11 | 12 | 12 | 1 |
| 12 | 4 | 4 | 1 |
| 13 | 14 | 7 | 2 |
| 14 | 6 | 6 | 1 |
| 15 | 12 | 12 | 1 |
CONCLUSIONS
In this article, we studied the length of the novel sequence – squared Pell sequence when considered modulo . We also introduced the ‘blocks’ within the period of this sequence and shown that length of any one period of the squared Pell sequence always contains either 1 or 2 blocks.
Conflict of Interest:
The authors confirm that this article contents have no conflict of interest to declare for this publication.
ACKNOWLEDGEMENTS
The authors are grateful to the referees for their valuable suggestions and support This research was conducted without any external funding. Therefore, I would like to respectfully request a waiver for the fee related to my research paper.
REFERENCES
Journal Articles:
- Horadam, A. F. (1971): Pell identities, Fib. Quart., 9 (3), 245 – 252, 263.
- Kramer J., Hoggatt V. E. Jr. (1972): Special cases of Fibonacci Periodicity, Fibonacci Quarterly,1 (5), 519 – 522.
- Ömür Deveci, Erdal Karaduman (2015): The Pell Sequences in Finite Groups, Utilitas Mathematica, 96, 263 – 276.
- Patel Rima P., Shah Devbhadra V. (2021): Blocks within the period of Lucas sequence, Ratio Mathematica, 41, 71 – 78.
- Patel Rima P., Shah Devbhadra V. (2017): Periodicity of generalized Lucas numbers and the length of its period under modulo , The Mathematics Today, 33, 67 – 74.
- Wall D. D. (1960): Fibonacci series modulo . The American Mathematical Monthly, 67, 525 – 532.
Books:
- Koshy Thomas (2001): Fibonacci and Lucas Numbers with Applications. John Wiley and Sons, Inc., New York.
- Koshy Thomas (2014): Pell and Pell-Lucas Numbers with Applications, Springer Science & Business Media, New York.
Thesis:
- Renault Marc (1996): Properties of the Fibonacci sequence under various moduli, Master’s Thesis, Wake Forest University.