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Carlitz Module Analogues of Mersenne Primes, Wieferich Primes, and Certain Prime Elements in Cyclotomic Function fields

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Carlitz Module Analogues of Mersenne Primes, Wieferich Primes, and Certain Prime Elements in Cyclotomic Function fields Journal of Number Theory 145 (2014) 181–193 Contents lists available at ScienceDirect Journal of Number Theory www.elsevier.com/locate/jnt Carlitz module analogues of Mersenne primes, Wieferich primes, and certain prime elements in cyclotomic function fields Nguyen Ngoc Dong Quan Department of Mathematics, University of British Columbia, Vancouver, British Columbia, V6T 1Z2, Canada a r t i c l e i n f o a b s t r a c t Article history: In this paper, we introduce a Carlitz module analogue Received 6 August 2013 of Mersenne primes, and prove Carlitz module analogues Received in revised form 10 April of several classical results concerning Mersenne primes. In 2014 contrast to the classical case, we can show that there are Accepted 6 May 2014 infinitely many composite Mersenne numbers. We also study Available online 7 July 2014 Communicated by Dinesh S. Thakur the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module context that Keywords: were first introduced by Dinesh Thakur. Carlitz modules © 2014 Elsevier Inc. All rights reserved. Cyclotomic function fields Mersenne primes Wieferich primes Contents 1. Introduction...................................................... 182 1.1. Notation.................................................... 183 2. A Carlitz module analogue of Mersenne primes.............................. 184 3. Wieferich primes and non-Wieferich primes................................. 187 4. The Carlitz annihilators of primes....................................... 189 E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jnt.2014.05.012 0022-314X/© 2014 Elsevier Inc. All rights reserved. 182 N.N. Dong Quan / Journal of Number Theory 145 (2014) 181–193 5. A criterion for determining whether a Mersenne number is prime.................. 191 Acknowledgments....................................................... 192 References............................................................ 192 Further reading......................................................... 193 1. Introduction In the number field context, a prime M is called a Mersenne prime if it is of the form M =2p − 1for some prime p. The Mersenne primes are among the integers of the form (1 + a)m − 1, where a, m are positive integers. It is a classical result that if (1 + a)m − 1 is a prime for some positive integers a, m with m ≥ 2, then it is necessary that a =1 and m = p for some prime p. There are many strong analogies between number fields and function fields. We refer the reader to the excellent references [2,10,12] for these analogies. The analogous pictures between number fields and function fields are clearly reflected when one considers the analogies between the two couples (Z, Q)and (F[T ], F(T )), where F is a finite field. The aim of this article is to search for new analogous phenomena between number fields and function fields. Specifically we will study the notion of Mersenne primes in the Carlitz module context, and relate them to the arithmetic of cyclotomic function fields. We also study the acquaintances of Mersenne primes including Wieferich and non-Wieferich primes in the Carlitz module setting that were introduced by Thakur [11,13]. Let us now introduce a Carlitz analogue of Mersenne primes. We begin by introducing some basic notation used here. s Let q = p , where p is a prime and s is a positive integer. Let Fq be the finite field of q q elements. Let A = Fq[T ], and let k = Fq(T ). Let τ be the mapping defined by τ(x) = x , and let kτ denote the twisted polynomial ring. Let C : A → kτ (a → Ca) be the Carlitz module given by CT = T + τ. Let R be a commutative k-algebra. The definition q of the Carlitz module C is equivalent to saying that CT (a) = Ta + a for every a ∈ R. m It is known that Cm(x)is analogous to (1 + x) − 1 ∈ Z[x]. This analogy suggests the following definition: a prime in A is called a Mersenne prime if it is of the form αCP (1), where P is a monic prime in A and α is a unit in A. To draw an analogy between the above notion of Mersenne primes and that of Mersenne primes in the number field context, we prove in Section 2 a Carlitz mod- ule analogue of the classical result in elementary number theory that was mentioned in the first paragraph of this introduction. Let us now describe the content of the paper. In Section 2, we introduce the notions of Mersenne numbers and Mersenne primes in the Carlitz module context. As remarked in [8], it is not known whether there are infinitely many primes p for which the Mersenne numbers 2p − 1are composite. In contrast to the number field setting, we prove in the Carlitz module context that for every q>2, there are infinitely many monic primes ℘ in Fq[T ]such that the Mersenne numbers C℘(1) are composite. N.N. Dong Quan / Journal of Number Theory 145 (2014) 181–193 183 In Section 3, we recall the notions of Wieferich and non-Wieferich primes in the Carlitz module context that were introduced by Dinesh Thakur [11,13]. Theorem 3.3 shows that every Mersenne prime is a non-Wieferich prime, which is analogous to a similar statement in the number field context. It is a classical result in elementary number theory that for a given odd prime p, p every prime q dividing the Mersenne number Mp := 2 − 1satisfies q ≡ 1 (mod p). The classical proof of this result is based on the notion of the order of an element modulo a prime. In Section 4, we prove a Carlitz module analogue of this result which states that for a given monic prime P in A, every monic prime Q dividing the Mersenne number ∈ F× ≡ MP := αCP (1) with α q satisfies Q 1 (mod P ). In order to prove this result, we introduce in Section 4 a notion of the Carlitz annihilator of a prime that is analogous to that of the order of an element modulo a prime. In the last section, using the arithmetic of cyclotomic function fields, we prove a criterion for determining whether a Mersenne number is prime. It is worth mentioning that Dinesh Thakur [13] found interesting relations linking Wieferich primes in the function field context to zeta values. 1.1. Notation In addition to the notation introduced before, let us fix some basic notation and definitions used throughout the paper. n Every nonzero element m ∈ A can be written in the form m = αnT +···+α1T +α0, where the αi are elements in Fq and αn =0. When m is of the form as above, we say that the degree of m is n. In notation, we write deg(m) = n. We use the standard convention that deg(0) = −∞. For each m ∈ A, define |m| := qdeg(m). Note that |m| is the number of elements of the finite ring A/mA. For basic properties of | ·|, we refer the reader to [10]. Fix an algebraic closure k¯ of k, and set Λ := λ ∈ k¯ Cm(λ) = 0 for some nonzero m ∈ A . ¯ For every nonzero element m ∈ A, define Λm := {λ ∈ k | Cm(λ) =0}. We recall the following definition. Definition 1.1. Let m ∈ A be a polynomial of positive degree. The field Km = k(Λm)is called a cyclotomic function field. For each polynomial m in A of positive degree, we define a primitive m-th root of Cm to be a root of Cm that generates the A-module Λm. Throughout the paper, for each m ∈ A of positive degree, we fix a primitive m-th root of Cm, and denote it by λm. Let Φm be the m-th cyclotomic polynomial, that is, the monic irreducible polynomial over k such that Φm(λm) =0. 184 N.N. Dong Quan / Journal of Number Theory 145 (2014) 181–193 For each polynomial m ∈ A of positive degree, set Sm := a ∈ A gcd(a, m)=1and0≤ deg(a) < deg(m) . (a) Fix an element m ∈ A of positive degree. For each element a ∈Sm, let σm be the (a) k-automorphism of Km defined by σm (λm) = Ca(λm). Let Gm denote the Galois group (a) of Km over k. It is well-known [5] that Gm := Gal(Km/k) = {σm | a ∈Sm}. 2. A Carlitz module analogue of Mersenne primes Let m ∈ A be a monic polynomial of degree d ≥ 1. By [10, Proposition 12.11], we can write Cm(x) ∈ A[x]in the form q q2 qd−1 qd Cm(x)=mx +[m, 1]x +[m, 2]x + ···+[m, d − 1]x + x ∈ A[x], (1) and hence C (x) 2 d−1 d m = m +[m, 1]xq−1 +[m, 2]xq −1 + ···+[m, d − 1]xq −1 + xq −1 ∈ A[x]. x It is well-known that Cm(x) = a|m Φa(x), where Φa(x) ∈ A[x]is the a-th cyclotomic a monic polynomial. If m = P for some monic irreducible polynomial P , then it is well-known [5] that ΦP (x) = CP (x)/x is an Eisenstein polynomial at P , i.e., the polynomial [m, i]is congru- ent to zero modulo P for each 1 ≤ i ≤ d − 1, and m is divisible by P but not divisible by P 2. From the discussion above, we see that Cm(x)is analogous to the polynomial (1 + x)m − 1 ∈ Z[x]in the classical cyclotomic theory. We now prove a lemma that naturally motivates the notions of Mersenne numbers and Mersenne primes in the Car- litz module setting. Lemma 2.1. Let m ∈ A be a monic polynomial of degree d ≥ 1, and let α ∈ A be a polynomial of degree h ≥ 0. Assume that q ≥ 3 and Cm(α) is a monic prime in A.
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