Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture

K. Anitha; I. Mumtaj Fathima; A. R. Vijayalakshmi

doi:10.1515/math-2022-0563

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Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture

K. Anitha , I. Mumtaj Fathima and A. R. Vijayalakshmi

Published/Copyright: March 24, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer k ≥ 2 , there are ≫ log x Lucas non-Wieferich primes p ≤ x such that p ≡ ± 1 ( mod k ) , assuming the a b c conjecture for number fields.

Keywords: abc conjecture; arithmetic progressions; Lucas sequences; Lucas non-Wieferich primes; Wieferich primes

MSC 2010: 11B25; 11B39; 11A41; 11N13

1 Introduction

Let a ≥ 2 be an integer. An odd rational prime p is said to be a Wieferich prime for base a if

a p − 1 ≡ 1 ( mod p 2 ) .

Otherwise, it is called a non-Wieferich prime for base a. In 1909, Wieferich [1] proved that if the first case of Fermat’s last theorem is not true for a prime p , then p is a Wieferich prime for base 2. Until today, 1093 and 3511 are the only known Wieferich primes for base 2. It is still unknown whether there are infinitely many Wieferich primes that exist or not, for any given base a . However, Silverman [2] proved that there are infinitely many non-Wieferich primes that exist for any base a , assuming the a b c conjecture (defined in Section 2.1). He established that for any fixed a ∈ Q ∗ , where Q ∗ = Q \ { 0 } and a ≠ ± 1 , if the a b c conjecture is true, then

∣ { primes p ≤ x : a p − 1 ≢ 1 ( mod p 2 ) } ∣ ≫ a log x .

In 2013, Graves and Ram Murty [3] extended Silverman’s result to certain arithmetic progressions. They showed that if a , k , and n are positive integers and assume the a b c conjecture, then

∣ { primes p ≤ x : p ≡ 1 ( mod k ) , a p − 1 ≢ 1 ( mod p 2 ) } ∣ ≫ a , k log x log log x .

Later, Chen and Ding [4] improved the lower bound to log x ( log log log x ) M log log x , where M is any fixed positive integer. Recently, Ding [5] further sharpened this bound to log x .

We prove a similar lower bound for non-Wieferich primes in Lucas sequences ( U n ( P , Q ) ) n ≥ 0 under the assumption of the a b c conjecture for number fields [6,7].

We now recall the definition of Lucas sequence of first kind.

Definition 1.1

[8] The Lucas sequence of first kind ( U n ( P , Q ) ) n ≥ 0 is defined by the recurrence relation,

U n ( P , Q ) = P U n − 1 ( P , Q ) − Q U n − 2 ( P , Q ) ,

for all n ≥ 2 with initial conditions U 0 ( P , Q ) = 0 and U 1 ( P , Q ) = 1 , where P and Q are non-zero integers with gcd ( P , Q ) = 1 .

Alternatively, we have the Binet formula

(1.1) U n ( P , Q ) = α n − β n α − β ,

where α and β are the zeros of the polynomial x 2 − P x + Q , following the convention that ∣ α ∣ > ∣ β ∣ .

Throughout this article, we simply write U n instead of U n ( P , Q ) if P and Q are fixed and we assume that the discriminant Δ ≔ P 2 − 4 Q is positive. In 2007, McIntosh and Roettger [8] defined Lucas non-Wieferich primes and studied their properties. We define Lucas non-Wieferich primes [9] as follows:

An odd prime p is called a Lucas–Wieferich prime associated to the pair ( P , Q ) if

U p − Δ p ≡ 0 ( mod p 2 ) ,

where Δ p denotes the Legendre symbol. Otherwise, it is called a Lucas non-Wieferich prime associated to the pair ( P , Q ) . We note that, if ( P , Q ) = ( 3 , 2 ) , then Δ = 1 and U p − Δ p = 2 p − 1 − 1 . Every Wieferich prime is thus a Lucas–Wieferich prime associated to the pair ( 3 , 2 ) [8].

Earlier, Ribenboim [10] proved that there are infinitely many Lucas non-Wieferich primes (named as binary recurring sequences) under the assumption of the a b c conjecture. Recently, Rout [9] proved some lower bounds for the number of Lucas non-Wieferich primes p such that p ≡ 1 ( mod k ) . However, we have found some gaps in his proof.

More specifically, Rout used the classical result of cyclotomic polynomial Φ m ( x ) , i.e., if p ∣ Φ m ( a ) , then either p ∣ m or p ≡ 1 ( mod m ) [11]. This result is true for any rational integer a but not for any general algebraic integer. Furthermore, in Rout’s proof [9, page 8, line 6], he mixed up the rational integer a with the algebraic number α / β . This was noted by Wang and Ding [12] in their article. We now provide an example that illustrates how Rout’s proof fails when a is replaced by α / β .

Let α = ( 7 + 37 ) / 2 and β = ( 7 − 37 ) / 2 . For m = 6 , the cyclotomic polynomial Φ 6 ( α / β ) = ( 860 + 140 37 ) / 9 . For a prime p = 5 , 5 ∣ Φ 6 ( α / β ) . But 5 ∤ 6 and 5 ≢ 1 ( mod 6 ) .

In this article, we will fill those gaps and prove that there are ≫ log x Lucas non-Wieferich primes p such that p ≡ ± 1 ( mod k ) using Ding’s [5] proof techniques. To the best of our knowledge, our main theorem is the first result that addresses the problem of Lucas non-Wieferich primes in arithmetic progressions. More precisely, we prove the following:

Theorem 1.2

Let k ≥ 2 be a fixed integer and let n > 1 be any integer. Assuming the a b c conjecture for number fields, then

primes p ≤ x : p ≡ ± 1 ( mod k ) , U p − Δ p ≢ 0 ( mod p 2 ) ≫ α , k log x .

2 Preliminaries

In this section, we will briefly discuss some of the fundamental concepts required for the sequel.

2.1 The a b c conjecture

2.1.1 The a b c conjecture for integers (Oesterlé, Masser) [7]

Given any real number ε > 0 , there is a constant C ε such that for every triple of coprime positive integers a , b , and c satisfying a + b = c , we have

c < C ε ( rad ( a b c ) ) 1 + ε ,

where rad ( a b c ) = ∏ p ∣ a b c p .

We now recall the definition of Vinogradov symbol.

Definition 2.1

[6] Let f and g are two nonnegative functions. If f < c g for some positive constant c , then we write f ≪ g or g ≫ f . The symbol ≪ is called the Vinogradov symbol.

2.1.2 The generalized a b c conjecture for algebraic number fields [6,7]

Let K be an algebraic number field with a ring of integers O K , and let K ∗ = K \ { 0 } . Let V K be the set of all primes on K , i.e., any υ ∈ V K is an equivalence class of non-trivial norms on K (finite or infinite). For υ ∈ V K , we define an absolute value ‖ ⋅ ‖ υ by

‖ x ‖ υ = ∣ ψ ( x ) ∣ if υ is infinite, corresponding to the real embedding ψ : K → C , ∣ ψ ( x ) ∣ 2 if υ is infinite, corresponding to the complex embedding ψ : K → C , N ( p ) − ord p ( x ) if υ is finite and p is the corresponding prime ideal ,

for all x ∈ K ∗ . For x ≠ 0 , ord p ( x ) denotes the exponent of p in the prime ideal factorization of the principal fractional ideal ( x ) .

For any ( a , b , c ) ∈ ( K ∗ ) 3 , the height is

H K ( a , b , c ) ≔ ∏ υ ∈ V K max ( ‖ a ‖ υ , ‖ b ‖ υ , ‖ c ‖ υ ) .

The radical of ( a , b , c ) ∈ ( K ∗ ) 3 is

rad K ( a , b , c ) ≔ ∏ p ∈ I K ( a , b , c ) N ( p ) ord p ( p ) ,

where p is the rational prime lying below the prime ideal p and I K ( a , b , c ) is the set of all prime ideals p of O K for which ‖ a ‖ υ , ‖ b ‖ υ , and ‖ c ‖ υ are not equal.

The a b c conjecture for an algebraic number field K states that for any ε > 0 ,

H K ( a , b , c ) ≪ ε , K ( rad K ( a , b , c ) ) 1 + ε

for all a , b , c ∈ K ∗ satisfying a + b + c = 0 .

2.2 Cyclotomic polynomial

We now recall the cyclotomic polynomial and some of its properties.

Definition 2.2

[11] Let m ≥ 1 be any integer. The mth cyclotomic polynomial is

Φ m ( X ) = ∏ h = 1 gcd ( h , m ) = 1 m ( X − ζ m h ) ,

where ζ m is the primitive mth root of unity.

It follows that

(2.1) X m − 1 = ∏ d ∣ m Φ d ( X ) .

Let P ( r ) denotes the greatest prime factor of r with the convention that P ( 0 ) = P ( ± 1 ) = 1 . The following lemma characterizes the prime divisors of Φ m ( α , β ) , where

Φ m ( α , β ) = ∏ h = 1 gcd ( h , m ) = 1 m ( α − ζ m h β ) .

Lemma 2.3

(Stewart [13, Lemma 2]) Let ( α + β ) 2 and α β be coprime non-zero integers with α / β not a root of unity. If m > 4 and m ≠ 6 , 12 then P ( m / gcd ( 3 , m ) ) divides Φ m ( α , β ) to at most the first power. All other prime factors of Φ m ( α , β ) are congruent to ± 1 ( mod m ) . Furthermore, if m > e 452 4 67 then Φ m ( α , β ) has at least one prime factor congruent to ± 1 ( mod m ) .

We remark that, Bilu et al. [14] reduced the lower bound e 452 4 67 to 30. In Lemma 2.3, Stewart [13] considered the cyclotomic polynomial

α m − β m = ∏ d ∣ m Φ d ( α , β ) .

But we take the prime divisors p of Φ m ( α / β ) such that p ∤ α β = Q . Hence, the prime divisors of Φ m ( α / β ) and Φ m ( α , β ) are the same. Thus, by using Lemma 2.3, the prime divisors of Φ m ( α / β ) are congruent to ± 1 ( mod m ) .

Lemma 2.4

(Rout [15, Lemma 2.10]) For any real number b with ∣ b ∣ > 1 , there exists a constant C > 0 such that

∣ Φ m ( b ) ∣ ≥ C ∣ b ∣ ϕ ( m ) ,

where ϕ ( m ) is Euler’s totient function.

2.3 Some lemmas

We need the following results for the proof of our main theorem.

Lemma 2.5

(Rout [9, Corollary 3.3]) Let p be a prime and coprime to 2 Q . Suppose U n ≡ 0 ( mod p ) and U n ≢ 0 ( mod p 2 ) . Then, U p − Δ p ≡ 0 ( mod p ) and U p − Δ p ≢ 0 ( mod p 2 ) .

That is, Lemma 2.5 says that if a prime p divides the square-free part of U n (defined below in Section 3), for some n ∈ N , then p is a Lucas non-Wieferich prime.

The rank of appearance (or apparition) of a positive integer k in the Lucas sequence ( U n ) n ≥ 0 is the least positive integer m such that k ∣ U m . We denote the rank of apparition of k by ω ( k ) if it exists [16]. In 1930, Lehmer [17] proved that a prime p divides U n if and only if ω ( p ) divides n . Thus, the prime p is a Lucas non-Wieferich prime if and only if ω ( p ) divides p − Δ p .

Lemma 2.6

(Rout [9, Lemma 3.4]) For sufficiently large n ≥ 0 , we have

(2.2) ∣ α ∣ n / 2 < ∣ U n ∣ ≤ 2 ∣ α ∣ n .

We recall the following lemma from [5].

Lemma 2.7

(Ding [5, Lemma 2.5]) For any given positive integer k, we have

∑ n ≤ x ϕ ( n k ) n k = c ( k ) x + O ( log x ) ,

where c ( k ) = ∏ p 1 − gcd ( p , k ) p 2 > 0 and the implied constant depends on k.

3 Main results

This section begins with definitions of square-free and powerful parts of an integer. For any positive integer n = ∏ i = 1 r p i a i , a i ≥ 1 , the square-free part of n is ∏ p i , where the product runs over all i with a i = 1 . The powerful part of n is ∏ p i a i , where the product runs over all i with a i ≥ 2 .

Let n > 1 be any integer and k ≥ 2 be any fixed integer. We write U n k = X n k Y n k , where X n k and Y n k are the square-free and powerful parts of U n k , respectively. Let us take X n k ′ = gcd ( X n k , Φ n k ( α / β ) ) and Y n k ′ = gcd ( Y n k , Φ n k ( α / β ) ) .

We note that by using Binet formula (1.1), we write

U n = β n α − β ( ( α / β ) n − 1 ) = β n Δ ( ( α / β ) n − 1 ) .

Thus,

(3.1) ( α / β ) n − 1 ∣ Δ U n .

We prove the following lemma, which is similar to the result in [9]. For the purpose of completeness, we present the proof here.

Lemma 3.1

Assume that the abc conjecture is true for the quadratic field Q ( Δ ) . Then, for any ε > 0 , we have

∣ X n k ′ ∣ ∣ Q ∣ ϕ ( n k ) ≫ ε ∣ U ϕ ( k ) ∣ 2 ( ϕ ( n ) − ε ) .

Proof

By the Binet formula (1.1), we have

(3.2) Δ U n k − α n k + β n k = 0 .

Applying the a b c conjecture for the number field K = Q ( Δ ) to equation (3.2) provides that:

for any ε > 0 , there exists a constant C ε such that

(3.3) H ( Δ U n k , − α n k , β n k ) ≤ C ε ( rad ( Δ U n k , − α n k , β n k ) ) 1 + ε ,

where

(3.4) rad ( Δ U n k , − α n k , β n k ) = ∏ p ∣ Q Δ U n k N ( p ) ord p ( p ) ≤ Q 2 Δ X n k 2 Y n k

and

(3.5) H ( Δ U n k , − α n k , β n k ) = max { ∣ Δ U n k ∣ , ∣ − α n k ∣ , ∣ β n k ∣ } ⋅ max { ∣ − Δ U n k ∣ , ∣ α n k ∣ , ∣ − β n k ∣ } ≥ ∣ Δ U n k ∣ ∣ − Δ U n k ∣ = Δ U n k 2 = Δ X n k 2 Y n k 2 .

Substituting (3.4) and (3.5) in (3.3), we obtain

(3.6) Y n k ≪ ε U n k 2 ε .

By using equation (2.1), we write

Φ n k ( α / β ) = ( α / β ) n k − 1 ∏ d ∣ n k Φ d ( α / β ) .

It follows that

Φ n k ( α / β ) ∣ U n k α n k − 1 .

Since U n k = X n k Y n k ,

Φ n k ( α / β ) ∣ X n k Y n k α n k − 1 .

As gcd ( Φ n k ( α / β ) , α ) = 1 , we have Φ n k ( α / β ) ∣ X n k Y n k . Since gcd ( X n k , Y n k ) = 1 , we obtain Φ n k ( α / β ) divides X n k or Φ n k ( α / β ) divides Y n k .

Suppose Φ n k ( α / β ) divides X n k , we have X n k ′ = gcd ( X n k , Φ n k ( α / β ) ) = Φ n k ( α / β ) and Y n k ′ = gcd ( Y n k , Φ n k ( α / β ) ) = 1 . Similarly, if Φ n k ( α / β ) divides Y n k , we obtain X n k ′ = 1 and Y n k ′ = Φ n k ( α / β ) . Thus, in either case, we obtain

X n k ′ Y n k ′ = Φ n k ( α / β ) .

By Lemma 2.4, we write

(3.7) ∣ X n k ′ Y n k ′ ∣ = ∣ Φ n k ( α / β ) ∣ ≥ C ∣ α / β ∣ ϕ ( n k ) = C ∣ α 2 / Q ∣ ϕ ( n k ) .

Hence, from equations (3.6), (3.7), and (2.2), we obtain

∣ X n k ′ U n k 2 ε ∣ ≫ ε ∣ X n k ′ Y n k ∣ ≥ ∣ X n k ′ Y n k ′ ∣ ≫ ε 1 ∣ Q ∣ ϕ ( n k ) ∣ α ∣ 2 ϕ ( n k ) ≫ ε 1 ∣ Q ∣ ϕ ( n k ) ∣ U ϕ ( k ) ∣ 2 ϕ ( n ) .

Therefore,

∣ X n k ′ ∣ ≫ ε 1 ∣ Q ∣ ϕ ( n k ) U ϕ ( k ) 2 ϕ ( n ) U n k 2 ε = 1 ∣ Q ∣ ϕ ( n k ) U ϕ ( k ) 2 ε U ϕ ( k ) 2 ( ϕ ( n ) − ε ) U n k 2 ε ≫ ε 1 ∣ Q ∣ ϕ ( n k ) ∣ U ϕ ( k ) ∣ 2 ( ϕ ( n ) − ε ) .

This completes the proof of the lemma.□

The following lemma is inspired by the result in [12, Lemma 2.4].

Lemma 3.2

If m < n , then gcd ( X m ′ , X n ′ ) = 1 or a power of Δ .

Proof

We suppose that gcd ( X m ′ , X n ′ ) > 1 is not a power of Δ . Let γ ( ≠ Δ ) be a prime element of Q ( Δ ) such that γ ∣ X m ′ and γ ∣ X n ′ . By the definitions of X m ′ and X n ′ , we write γ ∣ Φ m ( α / β ) and γ ∣ Φ n ( α / β ) .

Since Φ m ( α / β ) ∣ ( α / β ) m − 1 and Φ n ( α / β ) ∣ ( α / β ) n − 1 , we obtain γ ∣ ( α / β ) m − 1 and γ ∣ ( α / β ) n − 1 . Thus, γ ∣ ( α / β ) gcd ( m , n ) − 1 .

For m < n , gcd ( m , n ) < n . Hence,

( α / β ) n − 1 = ( α / β ) n − 1 ( α / β ) gcd ( m , n ) − 1 ( α / β ) gcd ( m , n ) − 1 .

As gcd ( Φ n ( α / β ) , ( α / β ) gcd ( m , n ) − 1 ) = 1 , we obtain

Φ n ( α / β ) ∣ ( α / β ) n − 1 ( α / β ) gcd ( m , n ) − 1 .

It follows that γ 2 ∣ ( α / β ) n − 1 . Thus, by using (3.1), we obtain γ 2 ∣ Δ U n . As γ ≠ Δ , γ 2 ∣ U n . Let p be a rational prime such that γ ∣ p . Since X n and U n are both integers and γ ∣ X n ′ , X n ′ ∣ X n . As a result, p ∣ X n and p 2 ∣ U n . This contradicts the definition of X n .□

The following lemma is an analogous result of [5, Lemma 2.6].

Lemma 3.3

Let S = { n : ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ > n k } and S ( x ) = ∣ S ∩ [ 1 , x ] ∣ . Then,

S ( x ) ≫ α , k x .

Proof

Let T = { n : ϕ ( n k ) > 2 c ( k ) n k / 3 } and T ( x ) = ∣ T ∩ [ 1 , x ] ∣ . By using equations (2.2) and (3.6), we write

(3.8) ∣ Y n k ′ ∣ ≤ ∣ Y n k ∣ ≪ ε ∣ U n k ∣ 2 ε ≤ 2 ∣ α n k ∣ 2 ε .

Substituting (3.8) in (3.7), we obtain

(3.9) ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ ≫ ε ∣ α ∣ 2 ( ϕ ( n k ) − ε n k ) .

By taking ε = c ( k ) / 3 in equation (3.9), we obtain

(3.10) ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ ≫ k ∣ α ∣ 2 ( ϕ ( n k ) − c ( k ) n k / 3 ) .

For any n ∈ T , (3.10) becomes,

∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ ≫ k ∣ α ∣ 2 ( ϕ ( n k ) − c ( k ) n k / 3 ) > ∣ α ∣ 2 c ( k ) n k / 3 ≫ α , k ∣ α ∣ 2 c ( k ) n k / 3 − log n k / log ∣ α ∣ n k > n k .

Hence, there exists an integer n 0 depending only on α , k such that if n ≥ n 0 and n ∈ T , then ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ > n k . Now, we write

S ( x ) = ∑ n ≤ x ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ > n k 1 ≥ ∑ n ≤ x n ≥ n 0 n ∈ T 1 = ∑ n ≤ x n ≥ n 0 ϕ ( n k ) > 2 c ( k ) n k / 3 1 .

and we note that

(3.11) ∑ n ≤ x ϕ ( n k ) ≤ 2 c ( k ) n k / 3 ϕ ( n k ) n k ≤ ∑ n ≤ x ϕ ( n k ) ≤ 2 c ( k ) n k / 3 2 c ( k ) 3 ≤ 2 c ( k ) 3 x .

Hence, by Lemma 2.7 and equation (3.11), S ( x ) becomes

S ( x ) ≥ ∑ n ≤ x n ≥ n 0 ϕ ( n k ) > 2 c ( k ) n k / 3 1 ≫ ∑ n ≤ x ϕ ( n k ) > 2 c ( k ) n k / 3 1 ≥ ∑ n ≤ x ϕ ( n k ) > 2 c ( k ) n k / 3 ϕ ( n k ) n k = ∑ n ≤ x ϕ ( n k ) n k − ∑ n ≤ x ϕ ( n k ) ≤ 2 c ( k ) n k / 3 ϕ ( n k ) n k ≥ c ( k ) x + O ( log x ) − 2 c ( k ) 3 x ≫ α , k x .

This completes the proof of Lemma 3.3.□

3.1 Proof of Theorem 1.2

For any n ∈ S , there exists a prime p n such that p n ∣ X n k ′ and p n ∤ n k . Since p n ∣ X n k ′ and X n k ′ ∣ X n k , we observe that p n ∣ U n k and p n 2 ∤ U n k . As P, Q, and Δ are fixed integers, it is clear that p n ∤ P Q Δ , except possibly finitely many primes. Hence, by using Lemma 2.5, we obtain

U p n − Δ p n ≢ 0 ( mod p n 2 ) .

Since p n ∣ Φ n k ( α / β ) , p n ∤ n k , and we use Lemma 2.3, we obtain p n ≡ ± 1 ( mod n k ) . Hence, for any n ∈ S , there is a prime p n satisfying

U p n − Δ p n ≢ 0 ( mod p n 2 ) , p n ≡ ± 1 ( mod n k ) .

From Lemma 3.2, we conclude that each p n ( n ∈ S ) is a distinct prime. Thus, we explore that

primes p ≤ x : p ≡ ± 1 ( mod k ) , U p − Δ p ≢ 0 ( mod p 2 ) ≥ ∣ { n : n ∈ S , ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ ≤ x } ∣ .

Since ∣ Q ∣ = ∣ α β ∣ , we write ∣ Q ∣ ϕ ( n k ) < ∣ α ∣ 2 n k , and we also have ∣ X n k ′ ∣ ≤ ∣ X n k ∣ ≤ ∣ U n k ∣ ≤ 2 ∣ α ∣ n k . Therefore, we obtain ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ < 2 ∣ α ∣ 3 n k . We now obtain

∣ { n : n ∈ S , ∣ Q ∣ ϕ ( n k ) ∣ X n k ′ ∣ ≤ x } ∣ ≥ ∣ { n : n ∈ S , 2 ∣ α ∣ 3 n k ≤ x } ∣ = n : n ∈ S , n ≤ log x / 2 3 k log ∣ α ∣ = S log x / 2 3 k log ∣ α ∣ .

Hence, by Lemma 3.3, we write

primes p ≤ x : p ≡ ± 1 ( mod k ) , U p − Δ p ≢ 0 ( mod p 2 ) ≥ S log x / 2 3 k log ∣ α ∣ ≫ α , k log x / 2 ≫ α , k log x .

This completes the proof.

Acknowledgments

The authors would like to thank the referees for their valuable comments and suggestions, which greatly improved the quality and presentation of this article. The second author, I. Mumtaj Fathima, would like to express her gratitude to the Maulana Azad National Fellowship for minority students, UGC.

Funding information: This research work was supported by a grant (MANF-2015-17-TAM-56982) from the University Grants Commission (UGC), Government of India.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2022-08-16

Revised: 2023-01-14

Accepted: 2023-02-01

Published Online: 2023-03-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0563

Keywords for this article

abc conjecture; arithmetic progressions; Lucas sequences; Lucas non-Wieferich primes; Wieferich primes

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