A note on balancing non-Wieferich primes in arithmetic progression
aa r X i v : . [ m a t h . N T ] J a n BALANCING NON-WIEFERICH PRIMES IN ARITHMETIC PROGRESSION
K. ANITHA, I. MUMTAJ FATHIMA, AND A R VIJAYALAKSHMIA
BSTRACT . A prime p is called balancing non-Wieferich prime if B p − ( p ) mod p ) ,where B n be the n -th balancing number and (cid:18) p (cid:19) denotes the Jacobi symbol. Under theassumption of the abc conjecture for the number field Q [ √ , S. S. Rout proved that thereare at least O (log x/ log log x ) said primes p ≡ mod r ) , where r > be any fixed integer.In this paper, we improve the lower bound such that for any given integer r > thereare ≫ log x primes p ≤ x satisfies B p − ( p ) mod p ) and p ≡ mod r ) , under the abc conjecture for the number field Q [ √ . This improves the recent result of Y. Wang and Y.Ding by adding an additional condition that the primes p are in arithmetic progression.
1. I
NTRODUCTION
Let b be an integer with b ≥ , we say an odd rational prime p is called a Wieferichprime for the base b if b p − ≡ mod p ) . A Wieferich prime for the base is simply calleda Wieferich prime. If the aforementioned congruence is not satisfies for an odd prime p ,it is called a non-Wieferich prime . That is p satisfies, p − mod p ) , then there are nointeger solutions to the first case of Fermat last theorem x p + y p = z p and x, y, z are notdivisible by p (see [13]). The known Wieferich primes are and . According tothe PrimeGrid project [7], these are the only Wieferich primes less than × . Also itis not known that, whether there are infinitely many Wieferich primes or non-Wieferichprimes for the given base b .Similarly, we can consider analogous results on balancing Wieferich and non-Wieferichprimes respectively. So our first priority is on balancing numbers, propounded by A.Behera and G. K. Panda [1]. Balancing number can be defined as, it is a positive integer n , which is a solution of the equation, . . . + ( n −
1) = ( n + 1) + ( n + 2) + . . . + ( n + k ) , where k ∈ Z + is called balancer to the balancing number n . In other words, a positiveinteger n is a balancing number if and only if n is a triangular number, that is n + 1 is a perfect square. Balancing number can be written in the form of recurrence relation, B n +1 = 6 B n − B n − for n ≥ with initial conditions B = 0 , B = 1 and the Binet Mathematics Subject Classification.
Key words and phrases. abc conjecture, balancing numbers, arithmetic progression, Wieferich primes,balancing Wieferich primes. formula for balancing number is B n = γ n − δ n γ − δ , where γ = 3 + 2 √ and δ = 3 − √ (see [1]). In [6] G.K. Panda and S. S. Rout discussedthe periodicity of balancing number modulo any natural number, which helps to findthe divisibility conditions of these numbers. They are also conjectured that there arethree primes , , and such that periods of balancing sequence modulo thesaid primes are equal to the periods modulo its square.Moreover, for any prime p > , B p − ( p ) ≡ mod p ) , where (cid:18) p (cid:19) denotes the Jacobisymbol [6]. Then S. S. Rout [9] called the prime p as a balancing Wieferich prime if itsatisfies the congruence B p − ( p ) ≡ mod p ) . Otherwise it is called balancing non-Wieferich prime and he proved that for an inte-ger r ≥ , under the assumption of the abc conjecture for the number field Q [ √ ,there are at least O (log x/ log log x ) balancing non-Wieferich primes p with p ≡ mod r ) (see [9]). Then U. K. Dutta et al. improved the lower bound from log x/ log log x to (log x/ log log x )(log log log x ) M , where M is any fixed positive integer. Recently, Y. Wangand Y. Ding [12] improved this lower bound to log x , but they excluded the conditionthat the balancing non-Wieferich primes p are in arithmetic progression.In this paper, we improve the lower bound to at least log x primes p ≤ x such thatbalancing non-Wieferich primes p are in arithmetic progression, under the assumptionof the abc conjecture. We here adopt the generalization of abc conjecture for the numberfield Q [ √ and work in Z [ √ which is the ring of integers of the number field Q [ √ .2. P RELIMINARIES AND SOME LEMMAS
We start this section with abc conjecture, it propounded by D. Masser ( ) and J.Oesterl´e ( ) and it states that let a, b, c are positive co prime integers satisfy a + b = c ,for any ǫ > max {| a | , | b | , | c |} ≪ ǫ ( rad ( abc )) ǫ , where rad ( abc ) = Y p | abc p .Now the analogue of abc conjecture for the number fields are as follows, for moredetails interested reader may refer to ([11], [4]). Let K be an algebraic number field and V K be the set of primes on K , that is any υ ∈ V K is an equivalence class of non-trivialnorms on K (finite or infinite). Let k x k υ := N K/ Q ( p ) − υ p ( x ) , if υ is defined by a primeideal p of the ring of integers O K in K and υ p is the corresponding valuation, where N K/ Q is the absolute value norm. Let k x k υ := | ρ ( x ) | e for all non-conjugate embeddings ALANCING NON-WIEFERICH PRIMES IN ARITHMETIC PROGRESSION 3 ρ : K → C with e = 1 if ρ is real and e = 2 if ρ is complex. Then the height of any triple ( a, b, c ) ∈ K ∗ defined as, H K ( a, b, c ) := Y υ ∈ V K max ( k a k υ , k b k υ , k c k υ ) and the radical of ( a, b, c ) denoted as, rad K ( a, b, c ) := Y p ∈ I K ( a, b, c ) N K/ Q ( p ) υ p ( p ) , where p be the rational prime with p Z = p ∩ Z and I K ( a, b, c ) is the set of all primes p of O K for which k a k υ , k b k υ , k c k υ are not equal. Now the abc conjecture for algebraicnumber field states that, for any ǫ > ,H K ( a, b, c ) ≪ ǫ, K ( rad K ( a, b, c )) ǫ , for all a, b, c ∈ K ∗ satisfy a + b + c = 0 , the implied constant depends on K and ǫ . Nowwe need some help from cyclotomic polynomial, which we defined below. Definition 2.1.
The m − th cyclotomic polynomial can be defined as, for any integer m ≥ , Φ m ( Y ) = Y gcd ( h,m )=10 For all integers m ≥ and a ≥ , then Φ m ( a ) ≥ a φ ( m ) . We need the following result to prove our main theorems and it is appear in [8]. Proposition 2.3. If p | Φ m ( a ) , a ∈ N , then either p | m or p ≡ mod m ) . If we replace integer a with real number α in Lemma 2.2, a similar type of inequalityholds for real α , which is available in [9]. It is stated as follows. Lemma 2.4. For any real number α with | α | > , there exists C > such that | Φ m ( α ) | ≥ C. | α | φ ( m ) . The following important lemmas are found in [9]. K. ANITHA, I. MUMTAJ FATHIMA, AND A R VIJAYALAKSHMI Lemma 2.5. Suppose that B n factored into X n Y n , where X n and Y n are square free and powerfulpart of B n respectively. If p | X n , then B p − (cid:0) p (cid:1) mod p ) . Lemma 2.6. For n ≥ and γ = 3+2 √ , then the n -th balancing number satisfies the followinginequality. γ n − < B n < γ n . 3. M AIN R ESULTS The following theorem closely follows the Theorem 3.1 of [9]. Theorem 3.1. If abc conjecture for the number field Q [ √ is true and r ≥ , n > are coprime integers then there are infinitely many primes p such that, B p − ( p ) mod p ) p ≡ mod r ) Proof. Let n > be a square free integer, which is relatively prime to r ≥ . We can write, B nr = X nr Y nr , where X nr , Y nr are square free and powerful part of B nr respectively.From Binet formula for balancing numbers we have, B nr = γ nr − δ nr γ − δ , then it can be write as √ B nr − γ nr + δ nr = 0 . Now the abc conjecture for Q [ √ implies that, for any ǫ > there exists a constant C ǫ such that H (4 √ B nr , − γ nr , δ nr ) ≤ C ǫ ( rad (4 √ B nr , − γ nr , δ nr )) ǫ , (3.2)Since γ and δ are units, all prime factors depends only on √ B nr . Therefore we have, rad (4 √ B nr , − γ nr , δ nr ) = Y p | √ B nr N ( p ) (3.3) ≤ X nr Y nr (3.4)The height of the triple is H (4 √ B nr , − γ nr , δ nr ) = max {| √ B nr | , | − γ nr | , | δ nr |} . (3.5) max {| − √ B nr | , | + γ nr | , | − δ nr |} (3.6) ≥ | √ B nr | . | − √ B nr | = 32 B nr (3.7) = 32 X nr Y nr . (3.8) ALANCING NON-WIEFERICH PRIMES IN ARITHMETIC PROGRESSION 5 On substituting (3.4) and (3.8) in (3.2), we obtain X nr Y nr ≤ C ǫ (8 X nr Y nr ) ǫ Y nr ≤ C ǫ (8 X nr Y nr ) ǫ Y nr ≪ ǫ ( X nr Y nr ) ǫ . Therefore we have Y nr ≪ ǫ B ǫnr . (3.9)Now we take, X ′ nr = gcd ( X nr , Φ nr ( γ/δ )) ,Y ′ nr = gcd ( Y nr , Φ nr ( γ/δ )) , From the recursion formula of cyclotomic polynomial we can write, Φ nr ( γ/δ ) = B nr Φ ( γ/δ ) δ nr − Y h | nr Φ h ( γ/δ ) , it follows that, Φ nr ( γ/δ ) | B nr Φ ( γ/δ ) . That is, Φ nr ( γ/δ ) | X nr Y nr Φ ( γ/δ ) . As prime divisor of Φ ( γ/δ ) is √ and √ Φ nr ( γ/δ ) , we say that gcd (Φ ( γ/δ ) , Φ nr ( γ/δ )) =1 . Since gcd ( X nr , Y nr ) = 1 , we obtain either Φ nr ( γ/δ ) | X nr or Φ nr ( γ/δ ) | Y nr . We sup-pose that Φ nr ( γ/δ ) | X nr , which means X ′ nr = gcd ( X nr , Φ nr ( γ/δ )) = Φ nr ( γ/δ ) and Y ′ nr = gcd ( Y nr , Φ nr ( γ/δ )) = 1 . Similar argument for Φ nr ( γ/δ ) | Y nr implies that X ′ nr = 1 and Y ′ nr = Φ nr ( γ/δ ) . Any of these cases, we finally get X ′ nr Y ′ nr = Φ nr ( γ/δ ) . (3.10)By using Lemma (2.4) we write, | X ′ nr Y ′ nr | = | Φ nr ( γ/δ ) | (3.11) ≥ C. | γ/δ | φ ( nr ) (3.12) = C. | γ | φ ( nr ) . (3.13)Since { B nr } are sequence of positive integers and using Lemma (2.6) we get, X ′ nr Y ′ nr ≥ C. ( γ φ ( r ) ) φ ( n ) (3.14) > C.B φ ( n ) φ ( r ) . (3.15)Now by combining equation (3.9) with (3.14), we obtain X ′ nr B ǫnr ≫ X ′ nr Y nr ≫ X ′ nr Y ′ nr ≫ B φ ( n ) φ ( r ) X ′ nr ≫ B φ ( n ) φ ( r ) B ǫnr . K. ANITHA, I. MUMTAJ FATHIMA, AND A R VIJAYALAKSHMI After simplification we write, X ′ nr ≫ B φ ( n ) − ǫ ) φ ( r ) . We choose ǫ < / and we obtain, X ′ nr ≫ B φ ( n ) − φ ( r ) . As X ′ nr is a product of distinct primes in Q [ √ and when n → ∞ , then φ ( n ) → ∞ . Weget lim n →∞ { primes p : p | X ′ ir , i ≤ n } = ∞ . Since prime p divides X ′ nr (implies p | X nr ) and from equation (3.10) we can write p di-vides Φ nr ( γ/δ ) . Hence by Proposition (2.3) and Lemma (2.5) we conclude that there areinfinitely many primes p such that B p − ( p ) mod p ) p ≡ mod r ) . (cid:3) Theorem 3.16. Let r > and n > are positive integers and assume the abc conjecture for thenumber field Q [ √ is true. Then { primes p ≤ x : p ≡ mod r ) B p − ( p ) mod p ) }≫ log x We need the following lemmas to prove Theorem (3.16). We now define τ M be the setof all numbers has exactly M + 1 prime factors and δ M = M +1 Y i =1 (1 − p i ) , where p i ’s aredistinct primes. Lemma 3.17. Suppose that the abc conjecture for number field Q [ √ is true. Then there existsan integer n depending only on γ, r, M and gcd ( n, r ) = 1 such that if n ∈ τ M with n ≥ n , then X ′ nr ≫ nr .Proof. We define ǫ = δ M φ ( r )2 r . Now from equation (3.14) we have, X ′ nr Y ′ nr ≫ γ φ ( nr ) (3.18)By using Lemma (2.6) and equation (3.9) we obtain, Y ′ nr ≪ Y nr ≪ ǫ B ǫnr < ( γ nr ) ǫ (3.19)On substituting (3.19) into (3.18) we get, X ′ nr ≫ γ φ ( nr ) − ǫnr ) . Since n and r are relatively prime, we write φ ( nr ) − ǫnr = φ ( n ) φ ( r ) − ǫnr. ALANCING NON-WIEFERICH PRIMES IN ARITHMETIC PROGRESSION 7 We take n ∈ τ M and by the definition of δ M we have φ ( n ) = n.δ M . Thus φ ( nr ) − ǫnr = nδ M φ ( r ) − ǫnr = ǫnr. Hence we conclude that, X ′ nr ≫ γ ǫnr > B ǫnr ≫ nr. (cid:3) Lemma 3.20. If i < n , then gcd ( X ′ nr , X ′ ir ) = 1 , where r > .Proof. We assume that gcd ( X ′ nr , X ′ ir ) > , for i < n , that is there exists a prime p suchthat p | X ′ nr and p | X ′ ir . By the definitions of X ′ nr and X ′ ir , we say that p | B nr , p | B ir . So that p | gcd ( B nr , B ir ) . By the divisibility conditions of B n we write, gcd ( B nr , B ir ) = B gcd ( nr, ir ) (see [5]). Thus p | B gcd ( nr, ir ) . We write, B nr = B nr B gcd ( nr, ir ) B gcd ( nr, ir ) . Since Φ nr ( γ/δ ) | Φ ( γ/δ ) B nr , we obtain Φ nr ( γ/δ ) | Φ ( γ/δ ) B nr B gcd ( nr, ir ) B gcd ( nr, ir ) . Thus, Φ nr ( γ/δ ) | B nr B gcd ( nr, ir ) as gcd (Φ nr ( γ/δ ) , Φ ( γ/δ )) = 1 and gcd (Φ nr ( γ/δ ) , B gcd ( nr,ir ) ) = 1 . Since p | B gcd ( nr, ir ) , wearrive p | B nr , which contradicts our assumption on p | X ′ nr . Hence gcd ( X ′ nr , X ′ ir ) = 1 . (cid:3) The proof of the following lemma available in [2]. Lemma 3.21. For any given positive integer r , which is relatively prime to n such that X n ≤ x φ ( nr ) nr = d ( r ) x + O (log x ) , where d ( r ) = Y p (1 − gcd ( p, r ) p ) > and the implied constant depends on r . Let us take T = { n : X ′ nr > nr } and T ( x ) = | T ∩ [1 , x ] | . The following lemma closelyfollows Lemma 2.6 of [2]. Lemma 3.22. We have T ( x ) ≫ x, where the implied constant depends only on γ, r . K. ANITHA, I. MUMTAJ FATHIMA, AND A R VIJAYALAKSHMI Proof. Let R = (cid:26) n : φ ( nr ) > d ( r )3 nr (cid:27) and R ( x ) = | R ∩ [1 , x ] | . We take ǫ = d ( r ) / in X ′ nr ≫ γ φ ( nr ) − ǫnr ) and get X ′ nr ≫ γ φ ( nr ) − d ( r ) nr ) . For any n ∈ R , we have φ ( nr ) > d ( r ) nr .Therefore, X ′ nr ≫ γ φ ( nr ) − d ( r ) nr ) > γ d ( r ) nr/ > nr. So there exists a integer n depending only on γ, r such that if n > n and n ∈ R , then X ′ nr > nr . Hence we obtain, T ( x ) = X n ≤ xX ′ nr >nr ≥ X n ≤ xn ≥ n (3.23) = X n ≤ xn ≥ n φ ( nr ) > d ( r ) nr/ (3.24)Since we note that, X n ≤ xφ ( nr ) ≤ d ( r ) nr/ φ ( nr ) nr ≤ X n ≤ xφ ( nr ) ≤ d ( r ) nr/ d ( r )3 (3.25) ≤ d ( r )3 x (3.26)Hence by Lemma (3.17) and equation (3.24) we obtain, T ( x ) ≥ X n ≤ xn ≥ n φ ( nr ) > d ( r ) nr/ ≫ X n ≤ xφ ( nr ) > d ( r ) nr/ ≥ X n ≤ xφ ( nr ) > d ( r ) nr/ φ ( nr ) nr = X n ≤ x φ ( nr ) nr − X n ≤ xφ ( nr ) ≤ d ( r ) nr/ φ ( nr ) nr ≥ d ( r ) x + O (log x ) − d ( r )3 x ≫ x This completes the proof of Lemma (3.22). (cid:3) ALANCING NON-WIEFERICH PRIMES IN ARITHMETIC PROGRESSION 9 4. P ROOF OF T HEOREM (3.16)The main idea of this theorem is to count number of primes p such that p divides X ′ nr ≤ x . For any n ∈ T, from the definition of T there exists a prime p n such that p n | X ′ nr and p n nr . Since X ′ nr | X nr and p n | X ′ nr and by using Lemma (2.5) we obtain B p n − ( pn ) mod p n ) . We note that p n | X ′ nr , X ′ nr | Φ nr ( γ/δ ) and p n nr . Therefore p n ≡ mod nr ) , by Lemma(2.3). Thus for any n ∈ T , there is a prime p n satisfying, B p n − ( pn ) mod p n ) ,p n ≡ mod nr ) . By Lemma (3.20), we conclude that p n ( n ∈ T ) are distinct primes. Thus we find that, { primes p ≤ X ′ nr : p ≡ mod r ) B p − ( p ) mod p ) } ≥ { n : n ∈ T, X ′ nr ≤ x } Since X ′ nr ≤ X nr ≤ B nr < γ nr , we can write { n : n ∈ T, X ′ nr ≤ x } ≥ { n : n ∈ T, γ nr ≤ x } = { n : n ∈ T, n ≤ log xr log γ } = T (cid:18) log xr log γ (cid:19) Hence by Lemma (3.22) we conclude that, { primes p ≤ x : p ≡ mod r ) B p − ( p ) mod p ) } ≥ T (cid:18) log xr log γ (cid:19) ≫ log x A CKNOWLEDGMENT The author I. Mumtaj Fathima wish to thank Moulana Azad National Fellowshipfor minority students, UGC. This research work is supported by MANF-2015-17-TAM-56982, University Grants Commission (UGC), Government of India.R EFERENCES [1] A. Behera, G. K. Panda, On the square roots of triangular numbers, Fib. Quart. , (1999), 98– 105.[2] Y. Ding, Non-Wieferich primes under the abc conjecture, C. R. Acad. Sci. Paris. Ser. I, (2019), 483–486. [3] U. K. Dutta, B. K. Patel, P. K. Ray, Balancing non-Wieferich primes in arithmetic progressions, Proc.Math. Sci, , 21 (2019).[4] K. Gy˝ory, On the abc conjecture in algebraic number fields, Acta Arith. , (2008), 281–295.[5] G. K. Panda, Some fascinating properties of balancing numbers, Congr. Numer. , (2009), 185–189.[6] G. K. Panda, S. S. Rout, Periodicity of balancing numbers, Acta Math. Hung. , Problems in analytic number theory. Balancing non-Wieferich primes in arithmetic progression and abc conjecture, Proc. JapanAcad. Ser. A. , (2016), 112–116.[10] R. Thangadurai, A. Vatwani, The least prime congruent to one modulo n, Amer. Math. Monthly. , (2011), 737–742.[11] P. Vojta, Diophantine approximations and value distribution theory. Lecture notes in mathematics, 1239,Springer, Berlin, 1987.[12] Y. Wang, Y. Ding, A note on balancing non-Wieferich primes (Chinese), Journal of Anhui NormalUniversity, (2020), 129–133.[13] A. Wieferich, Zum letzten Fermatschen theorem (German), J. Reine Angew Math. , (1909), 293–302.D EPARTMENT OF M ATHEMATICS , SRM IST R AMAPURAM , C HENNAI NDIA R ESEARCH S CHOLAR , D EPARTMENT OF M ATHEMATICS , S RI V ENKATESWARA C OLLEGE OF E NGI - NEERING , A FFILIATED TO A NNA U NIVERSITY , S RIPERUMBUDUR , C HENNAI NDIA D EPARTMENT OF M ATHEMATICS , S RI V ENKATESWARA C OLLEGE OF E NGINEERING , S RIPERUMBUDUR ,C HENNAI NDIA Email address , K. Anitha: [email protected] Email address , I. Mumtaj Fathima: [email protected] Email address , A R Vijayalakshmi:, A R Vijayalakshmi: