Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I

: The aim of this work is to present a method using the cyclic sequences { M k } , { θ t,k } and { ψ t,k } in the finite fields F ρ , with ρ a prime, that yield divisors of Mersenne, Fermat and Lehmer numbers. The transformations τ t and σ t are introduced which lead to the proof of the cyclic nature of the sequences { θ t,k } and { ψ t,k } . Results on the roots of the H ( x ) -polynomials in F ρ form the central theme of the study.


Introduction
Fermat (1601-1665) considered a sequence of numbers of the form 2 m +1, where m is of the form 2 n .Since the few initial terms of this sequence yielded prime numbers successively, Fermat was under the impression that he had really obtained a formula for primes.It was Euler (1707-1783) who detected a flaw in the concept of Fermat when he found the divisibility of the number 2 32 + 1 by 641.Numbers of the form 2 2 n + 1 (n ≥ 0) are called the Fermat numbers.
Numbers of the form 2 n − 1 are referred to as Mersenne numbers and primes of this form are called Mersenne primes, named after Fr.Martin Mersenne (1588-1648).A necessary condition for the number 2 n − 1 to be a prime is that n shall be a prime (see, for e.g., Hardy and Wright [3]).Brillhart [1], Brillhart and Johnson [2], Kang [4], and Kravitz [5] have presented certain results on the divisors of Mersenne numbers.There is another type of numbers of interest.Numbers of the form 2 n + 1 are named after Lehmer, as per a reference by Ribenboim in [9].It turns out that Fermat numbers are particular cases of Lehmer numbers.Leyendekkers and Shannon [6] have brought out certain remarkable properties of Mersenne and Fermat numbers.
In the sequel, it is established that certain cyclic sequences in the finite fields F ρ , with ρ a prime, yield the divisors of Mersenne, Fermat and Lehmer numbers.The main results of this study are contained in Theorems 2.12, 2.14, 3.1, 3.4, 4.6, 5.5, Corollary 5.4, Theorems 6.1, 7.10, 8.2 and 8.3.
2 The sequences of polynomials over Z Notations.Let N and Z denote the sets of natural numbers and integers, respectively.
Problem of motivation.To settle the question concerning the common solutions of two Pell's equations, the concept of the characteristic number of two simultaneous Pell's equations was introduced by Mohanty and the author in [7].A generalized version of this method was presented by the author in [8].Two functions viz.a(t) and b(t) were introduced in [7] as follows: Let t be a natural number.Define a(t) = A 2 t−1 and b(t) = B 2 t−1 , where A r + B r √ D denotes a solution of the Pell's equation A 2 − DB 2 = 1, D being a square-free natural number.The properties possessed by these functions was the focus of attention in [8].These functions satisfy the relations a(t + 1) = 2(a(t)) 2 − 1 and b(t + 1) = 2a(t)b(t).The first relation implies 2a(t + 1) = (2a(t)) 2 − 2. This property was the motivating point for the present work.As a consequence, certain polynomial sequences in Z[x] are introduced in this section.Definition 2.1.(The polynomial sequence {F k (x)} in Z[x]).Define the infinite sequence {F k (x)} (k ≥ 1) in Z[x] as follows: (2.1) Then we have F 5 (x) = x 16 − 16x 14 + 104x 12 − 352x 10 + 660x 8 − 672x 6 + 336x 4 − 64x 2 + 2, etc.
By Eisenstein's criterion, it follows that the F k (x)'s are irreducible over Z for k ≥ 2.
Definition 2.2.(The polynomial sequences {G k (x)} and {H k (x)} in Z[x]).Define the infinite sequences {G k (x)} and {H k (x)} (k ≥ 0) over Z as follows: Equivalently we have (2.4) Definition 2.3 (Matrix of polynomials).We define a matrix with two rows contributed by the sequences {G k (x)} and {H k (x)} as follows:

Properties of the polynomial sequences
Several properties of the sequences under consideration can be established by the method of induction.We obtain some identities.

Determinants of sub-matrices of a(x)
We consider determinants of 2 × 2 sub-matrices of a(x).By induction, the following property possessed by the elements in any two successive columns of a(x) is got, using (2.3) and (2.4): Theorem 2.2.The following results hold: (2.10)

Inter relationships among the terms of the sequences
Theorem 2.3.For all integers k ≥ 1, we have: (2.12) Theorem 2.4.The following relationships hold for all integers k ≥ 0: )

Factorization results for polynomials
By repeated application of (2.3) and (2.4), we obtain Theorem 2.5 (Reduction formulae).For r < s, we have (2.21) Corollary 2.1.The following results hold: Theorem 2.6 (Factorization of polynomials).For all integers j ≥ 1, we have Applying the reduction formulae provided by Theorem 2.5, we obtain the following: Theorem 2.7.For all integers k, j ≥ 1, we have By induction, we have the following theorem.
Theorem 2.8 (Generalized result on the factorization of polynomials).For j ∈ N and k ≥ 0, (2.28) (2.29) Theorem 2.9.It holds that (2.33) Now we consider the question: Given j ∈ N , which are the polynomials in the G(x)-sequence (respectively, H(x)-sequence) not divisible by G j (x) (respectively, H j (x))?We obtain an important result.

Arithmetic progressions
Theorem 2.13.The sequences {G k (x)} and {H k (x)} of polynomials over Z contain infinite number of infinite sub-sequences, the subscripts of the terms of which are in arithmetic progression, with non-trivial common factors.
Example 2.1.We have Theorem 2.14.If 2m + 1 is a prime ≥ 5, then there do not exist G j (x) and H j (x) such that 0 < j < m and As a consequence of the foregoing discussion, it follows that the only possibilities of the divisors of the polynomials G m (x) and H m (x) are as provided by Theorem 2.12.

Products and quotients of polynomials
Proof.The leading coefficient and the coefficient of For j ∈ N and k ≥ 0, by Theorem 2.8 we have As a consequence of Theorem 2.15, we obtain the following result: Theorem 2.16.The quotient polynomials and

Satellite polynomials
Let us consider G m (x) and H m (x), where 2m + 1 is a composite number.The result contained in Theorem 2.16 leads to the following.
As a consequence of Theorems 2.12 and 2.16, we have Theorem 2.17.
) and at least one satellite polynomial.
3 The M -sequences and cycles in the field F ρ The polynomial sequences It would be worthwhile to determine the interplay among the values assumed by these sequences in a finite field.We deal with the first sequence in this section and the other two sequences will be taken up in the next section.
Let ρ be given odd prime.Consider the field Choose any element M ∈ F ρ and fix it.Define the infinite sequence {M k } in F ρ as follows: where F is defined by (2.1) and each M k is reduced modulo ρ.
We have Thus the terms of the sequence {M k } are polynomial expressions in M with coefficients from F ρ .The sequence {M k } will be referred to as the M -sequence in F ρ .
In the case of ρ = 3, the two sequences become identical.Let us assume ρ > 3 so that the two sequences are distinct.Definition 3.3 (Singular and non-singular sequences).We refer to the sequence This terminology is used in the sense that the latter sequence possesses a property in common with some other non-stationary M -sequence in F ρ as would be seen in the course of subsequent development of the theory.
If we start with M = 0, 2 or −2, we end up with the stationary M -sequence Hence, in order to obtain non-stationary M -sequences in F ρ , we have to exclude the values a necessary condition for getting a non-stationary M -sequence in F ρ is that M ̸ = 0, ±1, ±2 and M 2 ̸ = 2, 3.However, the restrictions M ̸ = 0, ±1, ±2 and M 2 ̸ = 2, 3 are not sufficient to produce a non-stationary M -sequence in F ρ as illustrated by the following.
Example 3.1.For ρ = 17 and M = 5, we get the sequence Theorem 3.1 (Necessary and sufficient condition).Given M ∈ F ρ , let M k be a general term of the M -sequence in F ρ .A necessary and sufficient condition for the M -sequence to be non-stationary is that M ̸ = 0, ±1, ±2 and We establish the sufficiency of the condition.Assume that M ̸ = 0, ±1, ±2 and Restriction on ρ: The conditions M ̸ = 0, ±1, ±2 and M 2 k ̸ = 2, 3, ∀ k ∈ N , imply that we have to choose ρ ≥ 11, in order to obtain a non-stationary M -sequence in F ρ .
Notation: Let ( p q ) denote the Jacobi symbol.In view of Theorem 3.1, if either of ( 2 ρ ) , ( 3 ρ ) is +1, then we have to exclude M ∈ F ρ with the property M 2 = 2 or 3, so as to get a non-stationary M -sequence.
If n is the smallest natural number such that M n+1 = M 1 then we say that the M -sequence has period n and the cycle has length n.
The following result lays the foundation for the major results in the later sections.Theorem 3.4 (Existence of M -cycle in F ρ ).If ρ is an odd prime ≥ 11, then there exists at least one M -cycle of length ≥ 2 in F ρ .Proof.We provide a construction proof.First consider the case when it follows that a r + 2 is expressible as a square in F ρ .Similarly, each one of a r−1 +2, a r−2 +2, . . ., a 1 +2 is a quadratic residue modulo ρ.However, since there exist quadratic non-residues modulo ρ, the predecessor elements in the above sequence cannot exhaust all the elements of F ρ .A similar observation applies to the sequence −1 → −1 → −1 → • • • .Take an element b 1 ∈ F ρ , not exhausted by the predecessor elements of the two stationary sequences in Since F ρ has exactly ρ elements, it follows that there exist integers k ̸ = j such that b k = b j .Thus the above sequence is non-stationary and it contains a cycle of length ≥ 2. A similar proof applies when only one or none of ( 2 ρ ), ( 3 ρ ) is +1.
Theorem 3.5 (Uniqueness of M -cycle and its length in F ρ ).For an M -cycle element in F ρ , (i) the M -cycle in which it occurs is unique.
(ii) the length of the M -cycle in which it occurs is unique.
4 The sequences {θ t,k } and {ψ t,k } in the field F ρ In this section, we introduce two sequences in the field F ρ and establish their properties.

M -cycle through a parameter
Let ρ be a given odd prime ≥ 11.In order to identify the relationship that an M -cycle has with the values assumed by the sequences {G k (x)} and {H k (x)} in F ρ , the introduction of a parameter becomes necessary.We consider a non-stationary M -cycle in F ρ attached with a parameter t.
Choose any in-cycle element M (t) ∈ F ρ .Then M (t) ̸ = 0, ±1, ±2.By our assumption, the resulting cycle in F ρ has a period n ≥ 2. Denote the cycle by where G and H are defined by (2.2), (2.3) or (2.4).Considering (2.5), we define the matrix a(M (t)) with elements from F ρ as follows: 4.2 Cyclic nature of the sequence {θ t,k } To establish the cyclic nature of the θ t,k -sequence, we require a relationship between M (t) and M (t − 1) through the terms of the θ t,k -sequence.This crucial relationship is presented below.
Theorem 4.1.The following relation holds in F ρ : Proof.The relation (4.4) holds for k = 0. Assume (4.4) for all positive integers up to k.In view of (2.11), we have Multiplying both sides by 2{M (t − 1)} 2 and using the relation we get Adding {M (t − 1) M (t)θ t,k } 2 to both sides, we get Using induction assumption, we obtain Hence the relation (4.4) follows by induction on k.
In order to derive the properties of the M (t)-cycles, we need two important transformations.The first to be introduced is the following.Definition 4.2 (The transformation τ t ).With respect to the M (t)-cycle under consideration, define τ t : F ρ → F ρ by the rule is the multiplicative inverse of M (t) in F ρ .
Theorem 4.2 (Cyclic nature of θ t,k as a function of t).The following relation holds for all k ≥ 0 : Proof.Follows from the relation (4.4).
Proof.If possible, suppose that there exists a natural number j such that θ t,j = 0. Using Theorem 4.2 successively, we get The second tool required is the following.
Corresponding to the cycle Theorem 4.6.Let t be varying.For each fixed k, the cyclic sequences {θ t,k (mod ρ)} and {ψ t,k (mod ρ)} as functions of t are periodic with the same period as that of the cycle Proof.Follows from the relations (2.3), (2.4), (4.1), (4.6) and (4.9).  5 Structure of {θ t,k } and {ψ t,k }-sequences in the field F ρ In this section, we exhibit how the θ t,k and ψ t,k -sequences, considered as functions of t, can be split into several identical parts and determine the structure of such parts in F ρ .Definition 5.1.(Neighboring elements and neighboring region).Let t be fixed.For θ t,k (respectively, ψ t,k ), we say that the neighboring elements are θ t,k−1 and θ t,k+1 (respectively, Ψ t,k−1 and ψ t,k ).For given t, any three consecutive elements in the θ t,k (respectively, ψ t,k )-sequence constitute a neighboring region with respect to the middle element among them.holds for some integer j, then θ t,j = 1 and ψ t,j+1 = ψ t,j + 2 (mod ρ). (5.2) Proof.Suppose (5.1) holds for some integer j.Using Theorem 2.2 and the relation (5.1) we have θ t,j (ψ t,j+1 − ψ t,j ) = 2.In view of the relation (2.6), we obtain ψ t,j+1 = ψ t,j + 2θ t,j .Consequently we get θ t,j = ±1.

Symmetric and skew-symmetric properties
Definition 5.2 (Subsets of a sequence with symmetric or skew-symmetric property).Consider two distinct sets with the same cardinality consisting of consecutive elements from a sequence {S n } (mod ρ).Let them be {S k , S k+1 , . . ., S k+r−1 } and {S h−r+1 , . . ., S h−1 , S h }.We impose the condition h ≥ k + 2r − 2 so that the first set can be referred to as the set in the left side and the second set can be referred to as the set in the right side.We say that the two sets possess symmetric property if (5.11) We say that the two sets have skew-symmetric property if (5.12) Using (2.3) and (2.4) we obtain the following theorem.
Theorem 5.3 (Extension of symmetric and skew-symmetric sets).Suppose there are two distinct pairs of consecutive elements in {θ t,k (mod ρ)}-sequence with symmetric property.Suppose the elements in the corresponding positions of {ψ t,k ( mod ρ)}-sequence possess skew-symmetric property.Then the cardinalities of these sets can be increased by 1, still maintaining the symmetric and skew-symmetric properties of the respective sets, with the inclusion of the successor elements in the forward movements of the left side sets and the predecessor elements in the backward movements of the right side sets.
By induction, we have the following corollary.
Corollary 5.2.Under the assumptions of Theorem 5.3, the cardinalities of the left side sets and the right side sets can be increased by any desired natural number, still maintaining the symmetric and skew-symmetric properties of the respective sets, with the inclusion of the successor elements in the forward movements of the left side sets and the predecessor elements in the backward movements of the right side sets.

Existence of identical parts
Because of the finiteness of F ρ , there exist two positive integers r > j such that θ t,r = θ t,j and ψ t,r = ψ t,j .
Similarly movements around θ t,r and ψ t,r yield the elements in the {θ t,k (mod ρ)} and {ψ t,k (mod ρ)}-sequences as in (5.15): The minimality of r implies that the sub-matrix succeeding the elements θ t,j and ψ t,j cannot be the one other than the sub-matrix preceding the elements θ t,r and ψ t,r.The elements have the property θ t,j−1 = θ t,r−1 , ψ t,j−1 = ψ t,r−1 , θ t,j = θ t,r , ψ t,j = ψ t,r and θ t,j+1 = θ t,r+1, , ψ t,j+1 = ψ t,r+1.Applying the forward and backward movements around θ t,j , ψ t,j , θ t,r , ψ t,r , we see that each sequence contains two identical parts constituted by the elements in (5.14) and (5.15).

Definition 5.3 (Compartments of a(M (t))
).The sub-matrix with 2 rows of a(M (t)), starting with 1 and ending with the next immediate 1 in the θ t,k (mod ρ)-sequence and starting with 1 and terminating with the next immediate − 1 in the ψ t,k (mod ρ)-sequence is called a compartment of the matrix a(M (t)).Given t, it follows that any two compartments in a(M (t)) have the same number of elements in the θ t,k -sequence as well as the ψ t,k -sequence.Let C 1 (t) denote the first compartment of a(M (t)).We call C 1 (t) the principal compartment in a(M (t)).
Theorem 5.4 (Nature of the compartment C 1 (t)).Given t, the number of elements of θ t,k and ψ t,k -sequences in the compartment C 1 (t) cannot be even.

Determination of the middlemost elements in the rows of C 1 (t)
Now we take up an important requirement in our study, namely the determination of the middlemost elements in the compartments.Around the middlemost elements in the rows of C 1 (t), we have This gives the result ψ t,ω = 0 (5.21) which plays a crucial role in the further development of our method of cyclic sequences.Applying Theorem 2.2 we obtain θ t,ω ψ t,ω+1 = 2.This implies ψ t,ω+1 ̸ = 0 and hence we have where is the multiplicative inverse of ψ t,ω+1 in F ρ .
5.7 Uniqueness of the middlemost entries in the rows of C 1 (t) One can establish that there does not exist a positive integer j ̸ = ω such that The proof is by contradiction, employing Theorem 2.2 and the method described earlier for the determination of the neighboring elements.Thus we obtain the following result.
Theorem 5.6.The middlemost positions in each compartment of a(M (t)) are occupied by the values of 2 M (t−1) and 0 in the first and second rows, respectively and these values are not attained at any other places in the concerned compartment.
The following distinguishing characteristic emerges: Remark 5.1.It is seen that Theorems 4.3 and 5.6 bring out the distinguishing feature of θ t,k (mod ρ) and ψ t,k (mod ρ)-sequences, namely that the θ t,k -sequence never attains the value of 0 whereas the ψ t,k -sequence attains zero exactly once in each compartment of a(M (t)).
Remark 5.2.Given M(t), it follows from equation (5.21) that ω is the smallest positive integer such that ψ t,ω attains the value of zero in F ρ .Definition 5.4 (Pivotal elements in C 1 (t)).The pair of middlemost entries in the first and second rows of C 1 (t) are referred to as the pivotal elements of C 1 (t).The middlemost position in the first or the second row of C 1 (t) is called the pivotal position of C 1 (t).
Theorem 5.7.For the M -cycle given by (4.1), all the C 1 (t)-compartments in the matrix a(t) have the same pair of pivotal elements, for all the positive integral values of t.
Proof.For a given t ∈ N , let the pivotal position of C 1 (t) in a(t) be ω.Then θ t,ω = 2 M (t−1) and ψ t,ω = 0.By Theorem 4.5, we have ψ t+1,ω = σ t (θ t,ω , ψ t,ω ) = 0. Since the ψ-sequence assumes the value of 0 only once in a compartment, it follows that the pivotal position of the compartment C 1 (t + 1) in a(t + 1) is also ω.Hence the theorem follows.
Corollary 5.3 (Transformation of the pivotal elements).The pivotal elements in C 1 (t) are transformed by τ t and σ t into the respective pivotal elements in C 1 (t + 1).
Proof.We have The theorem follows from (5.26) and (5.27).
Corollary 5.4 (Periodicity of the {θ t,k } and {ψ t,k }-sequences).Consider the cycle . For each t, the period of the cyclic sequence θ t,k (respectively, ψ t,k ) as a function of k is 2ω + 1.

Existence of roots of polynomials of H(x)-sequence in finite fields
The existence of a nontrivial M -cycle has been established in Theorem 3.4.Given M (t), we have seen in Section 5, how the θ t,k (mod ρ) and ψ t,k (mod ρ)-sequences, considered as functions of t, split into several identical parts.A remarkable property in the determination of the structure of such parts is provided by equation (5.21), viz. the existence of a least positive integer ω such that ψ t,ω = 0.This implies that M(t) satisfies the polynomial H ω (x).If ω = 1, then ψ t,1 = 0 implies M (t) = −1 from which we get the M -cycle −1 → −1 → −1 → . . .Thus M(t) contributes the root of the polynomial H 1 (x).Consider the case when ω > 1 so that we have n > 1.Take the cycle Hence a root α of the polynomial H d (x) in F ρ gives rise to an M (t)-cycle of length n such that each element of this cycle is a root of the polynomial H d (x).Thus we are led to an important result on the attainment of the roots of the H(x)-polynomial stated as follows: Theorem 6.1 (Necessary and sufficient condition).Let ρ be a given odd prime ≥ 11.An M (t)-cycle of length n exists in the field F ρ if and only if there exists a positive integer ω ≥ n such that the polynomial H ω (x) attains roots at n distinct elements of F ρ with pivotal position ω in the compartments C 1 (t), C 1 (t + 1), . . ., C 1 (t + n − 1) of the matrices a(M (t)), a(M (t + 1)), . . ., a(M (t + n − 1)), respectively.One can choose ω as the least positive integer with this property.Corollary 6.1.If α is a root of H ω (x) in F ρ , then α 2 − 2 is also a root of H ω (x).Corollary 6.2.Every root of an H(x)-polynomial occurring in F ρ is an element of a unique M -cycle.

A relationship involving the pivotal position in C 1 (t)
The concept of pivotal elements in a compartment was introduced in Section 5. Now we consider the identification of a relationship involving the pivotal position of the compartment C 1 (t).

Formation of new sequences
Let the parameter t be given.We form two new sequences as follows: η-sequence is formed with the sums of two consecutive terms of the ψ-sequence; ζ-sequence is formed from ψ-sequence by taking the terms from ψ t,ω onwards.We have the following definition.

Conclusion
In the material that has been hitherto presented, we have established how an M -cycle in the finite field F ρ yields the factors of Mersenne, Fermat and Lehmer numbers.It is pertinent to consider the converse result.A question that arises is how to find the M −cycles from the factors of given Mersenne, Fermat and Lehmer numbers.This will be taken up in a subsequent study.

Definition 3 . 6 (
In-cycle element).A cycle-forming element in F ρ is called an in-cycle element.Definition 3.7 (Ex-cycle element).A cycle-contributing element but not cycle-forming element in F ρ is called an ex-cycle element.Theorem 3.3.If M is an in-cycle or ex-cycle element in F ρ , then −M is an ex-cycle element.Definition 3.8 (M -cycle).If M is an in-cycle element in F ρ , then a cycle beginning with M and ending with M is called an M -cycle and the numbers in the cycle are called its elements.background prime for the M -cycle.Definition 3.10 (Length of an M −cycle).Consider an M -sequence (mod ρ) containing a cycle

Table 4 .
1 consisting of the terms of the sequences θ t,k and ψ t,k (mod 26879), using (2.2) or equivalently (2.3) and (2.4).The table consists of three parts contributed by the values of M ′ i s.As a consequence of the cyclic nature of θ t,k and ψ t,k as functions of t, starting with any one part of the above table, we can construct the other two parts of the table.