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≫ p p p 2 2 2 p α ) ) ) . , , p log ≤ snnWeeihprime, non-Wieferich is x x uhthat such a α as ≥ ∈ n n fixed any and 2 abc abc x Q x ∞ → × p conjecture. conjecture + α , a a ees- He . y p − p 6= . 1 = 6≡ p ± (1) (2) z a 1 p is 1 s - , , , 2 SRINIVAS KOTYADA AND SUBRAMANI MUTHUKRISHNAN

In this paper, we study non-Wieferich primes in algebraic number fields of class number one. More precisely, we prove the following Theorem 1.1. Let K = Q(√m) be a real quadratic field of class num- ber one and assume that the abc conjecture holds true in K. Then there are infinitely many non-Wieferich primes in K with respect to the unit ε satisfying ε > 1. O | | Theorem 1.2. Let K be any algebraic number field of class number one and assume that the abc conjecture holds true in K. Let η be a (j) (j) unit in K satisfying η > 1 and η < 1 for all j =1, where η is the jth conjugateO of η.| Then| there| exist| infinitely many6 non-Wieferich primes in K with respect to the base η.

The plan of this article is as follows. In section 2, we shall define the abc conjecture for number fields. In section 3, a brief introduction to Wieferich/non-Wieferich primes over number fields will be given and in section 4 and 5, we shall prove Theorem 1.1 and Theorem 1.2, respectively.

2. The abc-conjecture The abc-conjecture propounded by Oester´eand Masser (1985) states that given any δ > 0 and positive integers a, b, c such that a + b = c with (a, b) = 1, we have c (rad(abc))1+δ, ≪δ where rad(abc) := p|abc p. Q The abc conjecture has several applications, the reader may refer to [9], [3], [6], [7] for details. To state the analogue of abc-conjecture for number fields, we need some preparations, which we do below. The interested reader may refer to [9], [3] for more details.

Let K be an algebraic number field and let VK denote the set of primes on K, that is, any v in VK is an equivalence class of norm on K (finite or infinite). Let x := N (p)−vp(x), if v is a prime defined by the || ||v K/Q prime ideal p of the ring of integers K in K and vp is the corresponding O e valuation, where NK/Q is the absolute value norm. Let x v := g(x) for all non-conjugate embeddings g : K C with e = 1|| if|| g is real| and| e = 2 if g is complex. Define the height→ of any triple a, b, c K× as ∈ H (a, b, c) := max( a , b , c ), K || ||v || ||v || ||v vY∈VK NON-WIEFERICH PRIME 3 and the radical of (a, b, c) by

vp(p) radK(a, b, c) := NK/Q(p) ,

p ∈ IYK (a,b,c) where p is a rational prime with pZ = p Z and IK(a, b, c) is the set of all primes p of for which a , b ∩, c are not equal. OK || ||v || ||v || ||v The abc conjecture for algebraic number fields is stated as follows: For any δ > 0, we have H (a, b, c) (rad (a, b, c))1+δ, (3) K ≪δ,K K for all a, b, c, K× satisfying a+b+c = 0, the implied constant depends on K and δ.∈

3. Wieferich/non-Wieferich primes in number fields Let K be an algebraic number field and be its ring of integers. OK A prime π K is called Wieferich prime with respect to the base ε ∗ if ∈ O ∈ OK εN(π)−1 1 (mod π2), (4) ≡ where N(.) is the absolute value norm. If the congruence (4) does not hold for a prime π K , then it is called non-Wieferich prime to the base ε. ∈ O

Notation: In what follows, ε will denote a unit in K and we shall n O write ε 1 = unvn, where un is the square free part and vn is the − 2 squarefull part, i.e., if π vn then π vn. We shall denote absolute value norm on K by N. | |

4. Proof of theorem (1.1)

Let K = Q(√m),m> 0 be a real quadratic field and K be its ring of ∗ O integers. Let ε K be a unit with ε > 1. The results of Silverman [8], Ram Murty∈ and O Hester [2] elucidated| | in the introduction use a key lemma of Silverman (Lemma 3, [8]). We derive an analogue of Silverman’s lemma for number fields which will play a fundamental role in the proof of the main theorems. Lemma 4.1. Let K = Q(√m) be a real quadratic field of class number one. Let ε ∗ be a unit. If εn 1= u v , then every prime divisor ∈ OK − n n π of un is a non-Wieferich prime with respect to the base ε. Proof. The assumption that K has class number one allows us to write n the element ε 1 K as a product of primes uniquely. Accordingly, we shall write − ∈ O εn 1= u v − n n for n N. Then ∈ εn =1+ πw, (5) 4 SRINIVAS KOTYADA AND SUBRAMANI MUTHUKRISHNAN with π un and π and w are coprime. As π is a prime, we have N(π)= p or p2, p| is a rational prime. Case (1): Suppose N(π)= p. From equation (5), we get εn(p−1) 1+(p 1)πw 1 (mod π2). ≡ − 6≡ Case (2): Suppose N(π)= p2. Again from equation (5), we obtain 2 2 εn(p −1) = εn(N(π)−1) = (1+ πw)(p −1) 1+ πw(p2 1) 1 (mod π2). ≡ − 6≡ Thus in either case, ε(N(π)−1) 1 (mod π2), 6≡ and hence π is a non-Wieferich prime to the base ε. 

The above lemma shows that whenever a prime π divides un for some positive integer n, then π is a non-Wieferich prime with respect to the base ε. Thus, if we can show that the set N(u ): n N is { n ∈ } unbounded, then this will imply that the set π : π un, n N is an infinite set. Consequently, this establishes the{ fact| that there∈ }are infinitely many non-Wieferich primes in every real quadratic field of class number one with respect to the unit ε, with ε > 1. Therefore, we need only to show the following | | Lemma 4.2. Let Q(√m) be a real quadratic field of class number one. ∗ Let ε K be a unit with ε > 1. Then under abc-conjecture for number∈ fields, O the set N(u ):| |n N is unbounded. { n ∈ } Proof. Invoking the abc-conjecture (3) to the equation n ε =1+ unvn (6) yields 1+δ 1+δ εn N(p)vp(p) = N(p)vp(p) N(p)vp(p) (7) | |≪   p|Yunvn
pY|un pY|vn for some δ > 0. Here the implied constant depends on K and δ.

As vp(p) 2 for any prime ideal p lying above the rational prime p, we have ≤

N(p)vp(p) N(u )2. (8) ≤ n pY|un

For a prime ideal p vn, let ep be the largest exponent of p dividing vn, ep | n i.e., p vn. As vn is the square-full part of ε 1, we have ep 2. Hence, || − ≥

2vp(p) 2+ep (1) N(p) N(p) for all prime ideals p with vp(p)=2. 2vp(p) ≤ ep (2) N(p) N(p) for all prime ideals p with vp(p)=1. ≤ NON-WIEFERICH PRIME 5

Thus

N(p)2vp(p) N(p)2+ep(p) N(p)ep(p) ≤ pY|vn Yp|vn Yp|vn vp(p)=2 vp(p)=1 N(p)2 N(p)ep(p) N(p)ep(p) ≤ Yp|vn Yp|vn Yp|vn vp(p)=2 vp(p)=2 vp(p)=1 ′ N(p)2 N(p)ep(p) N(p)ep(p), ≤ Yp Yp|vn Yp|vn vp(p)=2 vp(p)=1 where ′ indicates that the product is over all primes p in such that OK vp(p) = 2. As it is well known that there are only finitely many ramified primes in a number field, it follows that the product is bounded by a constant A (say). Thus, we have

N(p)vp(p) AN(v ). (9) ≤ n pY|vn p Combining equations (7), (8) and (9), we get

1+δ n 2 ε N(un) N(vn) . (10) | |≪  p  Now, as ε > 1, | | N(u )N(v )= N(εn 1) 2 εn 1 < 2 ε n, n n − ≤ | − | | | i.e., N(v ) < 2 ε n/N(u ). n | | n Substituting the above expression in (10), we obtain

n/2 (1+δ) n 2 ε ε N(un) | | . | |≪  N(un) Thus, p 3(1+δ) n(1−δ) (N(u )) 2 ε 2 . n ≫| | Thus, for a fixed δ, N(un) as n . This proves the lemma and hence completes the proof→ ∞ of the theorem.→ ∞ 

5. Non-Wieferich primes in algebraic number fields In this section, we generalize the arguments of previous section to arbitrary number fields. From now onwards, K will always denote an algebraic number field of degree [K : Q] = l over Q of class number one. Let r1 and r2 be the number of real and non-conjugate complex embeddings of K into C respectively, so that l = r1 +2r2. We begin with an analogue of Lemma (4.1). 6 SRINIVAS KOTYADA AND SUBRAMANI MUTHUKRISHNAN

Lemma 5.1. Let ε be a unit in . If εn 1= u v , then every prime OK − n n divisor π of un is a non-Wieferich prime with respect to the base ε. Proof. Let N(π)= pk, where p is a rational prime and k is a positive integer. Then k k εn(N(π)−1) = εn(p −1) = (1+ wπ)(p −1) 1+(pk 1)wπ 1 (mod π2). ≡ − 6≡ This implies εN(π)−1 1 (mod π2). 6≡ Thus, the lemma shows that π is a non-Wieferich prime to the base ε whenever the hypothesis of the lemma is met. Now, under the abc conjecture for number fields, we show below the existence of infinitely many non-Wieferich primes.

Lemma 5.2. The set N(un): n N is unbounded, where un’s are as defined in Lemma (5.1){ . ∈ }

n Proof. By the hypothesis of the lemma, we have ε =1+ unvn, where n × ε , 1,unvn K . Applying the abc conjecture for number fields to the above equation,∈ we obtain max( u v , 1 , εn ) ( N(p)vp(p))1+δ, (11) | n n|v | |v | |v ≪ vY∈VK p|Yunvn for some δ > 0. Note that for the absolute value . in V , we have | | K εn max( u v , 1 , εn ). (12) | | ≤ | n n|v | |v | |v vY∈VK

As vp(p) l for any prime ideal p lying above the rational prime p, we have ≤ N(p)vp(p) N(u )l. (13) ≤ n pY|un

As before, we denote by ep the largest exponent of p which divides vn, ep i.e., p v . Clearly ep 2. Then || n ≥

N(p)2vp(p) N(p)2l+ep(p) N(p)ep(p) ≤ pY|vn Yp|vn Yp|vn vp(p)≥2 vp(p)=1 N(p)2l N(p)ep(p) N(p)ep(p) ≤ Yp|vn Yp|vn Yp|vn vp(p)≥2 vp(p)≥2 vp(p)=1 ′ N(p)2l N(p)ep(p) N(p)ep(p), ≤ Yp Yp|vn Yp|vn vp(p)≥2 vp(p)=1 where ′ indicates that the product is over all primes p in such that OK vp(p) 2. As there are only finitely many ramified primes in a number field,≥ it is bounded by a constant B (say). Thus, we have NON-WIEFERICH PRIME 7

N(p)vp(p) BN(v ). (14) ≤ n pY|vn p Therefore, the equations (11) - (14) yield 1+δ n l ε N(un) N(vn) . (15) | |≪  p  Note that in the case of real quadratic fields, the unit ε satisfies ε > 1 and this information was crucial in proving Theorem 1.1. However,| | in the case of general number fields, the following result (see Lemma 8.1.5, [1]) comes to our rescue. We state this result as

Lemma 5.3. Let E = k Z :1 k r1 + r2 . Let E = A B be a { ∈ ≤ ≤ } (k) ∪ proper partition of E. There exists a unit η K with η < 1, for k A and η(k) > 1, for k B. ∈ O | | ∈ | | ∈ Taking A = k : 1 < k r1 + r2 and B = 1 , Lemma 5.3 produces {∗ ≤ } (k) { } (k) a unit η K such that η > 1 and η < 1, where η denotes the kth conjugate∈ O of η,k = 1.| Since,| every| unit| satisfies (15), replacing ε with η in (15), we obtain6 1+δ n l η N(un) N(vn) , (16) | |≪  p  where, by abuse of notation, we shall denote ηn 1 = u v , with u − n n n and vn denoting the same quantities as defined earlier. Now, N(u )N(v )= N(ηn 1)=(ηn 1)(η(2)n 1)(η(3)n 1) (η(l)n 1). n n − − − − ··· − By Lemma 5.3, η(j)n 1 < 2 for all j, 2 j l. | − | ≤ ≤ Thus, N(u )N(v )

(2l−1)(1+δ) n 1−δ (N(u )) 2 η 2 . (17) n ≫| | For a fixed δ, the right hand side of (17) tends to as n . ∞ → ∞ Therefore the set N(un): n N is unbounded. This shows that there are infinitely{ many non-Wieferich∈ } primes in K with respect to the base η.  Acknowledgement We express our indebtedness to Prof. M. Ram Murty for initiating us into this project and for having many fruitful discussions. The second author would like to thank Prof. T.R. Ramadas for encouragement and also acknowledges him with thanks for the financial support extended by DST through his J.C. Bose Fellowship. Our sincere thanks to the referee for pointing out some errors and suggesting some changes in an 8 SRINIVAS KOTYADA AND SUBRAMANI MUTHUKRISHNAN earlier version of this paper. This work is part of the Ph D thesis of the second author. References [1] J. Esmonde, M. Ram Murty, Problems in Algebraic Number Theory, Grad- uate texts in Mathematics, Springer-Verlag New York, Inc. [2] H. Graves, M. Ram Murty, The abc conjecture and non-Wieferich primes in arithmetic progressions, J. Number Theory 133 (2013) 1809-1813. [3] K. Gy˝ory, On the abc conjecture in algebraic number fields, Acta Arith. 133 (2008), no. 3, 281-295. [4] D. A. Marcus, Number Fields, Graduate texts in mathematics, 1977, Springer-Verlag. [5] PrimeGrid Project, http://www.primegrid.com/. [6] A. Nitaj, The abc conjecture homepage (\protect\vrule width0pt\protect\href{http://www.math.unicaen.fr/\string~nitaj/abc.html [7] M. Ram Murty, The ABC conjecture and exponents of class groups of qua- dratic fields, Contemporary Mathematics 210 (1988), 85-95. [8] J. H. Silverman, Wieferich’s criterion and the abc-conjecture, J. Number Theory 30 (1988) no. 2, 226-237. [9] P. Vojta, Diophantine approximations and value distribution theory, Lecture Notes in mathematics (1980), Springer-Verlag. [10] A. Wieferich, Zum letzten Fermat’schen Theorem, J. Reine Angew. Math. 136 (1909), 293-302.

Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai 600 113, India

Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chen- nai 603 103, India E-mail address, Kotyada Srinivas: [email protected] E-mail address, Subramani Muthukrishnan: [email protected]