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Indian J. Pure Appl. Math., 46(5): 701-714, October 2015 c Indian National Science Academy ° DOI: 10.1007/s13226-015-0140-9 THE NUMERICAL FACTORS OF ∆n (f, g)1 Qingzhong Ji and Hourong Qin Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China, e-mails: [email protected], [email protected] (Received 19 June 2014; after final revision 30 September 2014; accepted 24 December 2014) Let α1 , α2 , . . . , αr be the roots of the polynomial f (x) = xr + a1 xr−1 + · · · + ar ∈ Z[x] and let g = {gn (X)}n∈N , where gn (X) = gn (x1 , x2 , . . . , xr ) ∈ Z[x1 , x2 , . . . , xr ] is a symmetric polynomial. For each n, put ∆n (f, g) = gn (α1 , α2 , . . . , αr ). In this paper, for a special symmetric polynomial sequence g, we investigate the numerical factors of ∆n (f, g). If p is a prime, we establish an analogue of Iwasawa’s theorem in algebraic number theory for the orders ordp (∆npt (f, g)) of the p-primary part of ∆npt (f, g) when t varies. Key words : Recurring series; Iwasawa theory; cyclotomic polynomial. 1. I NTRODUCTION Throughout this paper, let Q, Z and N denote the field of rational numbers, the ring of rational integers and the set of nonnegative integers, respectively. Let N∗ = N \ {0}. As usual, let ordp denote the p-adic valuation of Qp such that ordp (p) = 1. Let α1 , α2 , . . . , αr be the roots of the polynomial f (x) = xr + a1 xr−1 + · · · + ar−1 x + ar (1) whose coefficients are rational integers. Suppose g = {gn (X)}n∈N is a polynomial sequence, where gn (X) = gn (x1 , x2 , . . . , xr ) ∈ Z[x1 , x2 , . . . , xr ] is a symmetric polynomial in r variables. For each n, put ∆n (f, g) = gn (α1 , α2 , . . . , αr ). 1 (2) Supported by NSFC (Nos. 11171141, 11471154, 11271177), NSFJ (Nos. BK2010007, BK2010362), PAPD and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (No. 708044). 702 QINGZHONG JI AND HOURONG QIN It is clear that ∆n (f, g) ∈ Z. For example, if gn (x1 , x2 , . . . , xr ) = ∆n (f, g) = Qr i=1 (1 − xni ), then r Y (1 − αin ) ∈ Z. (3) i=1 This function was introduced by Pierce [4] who studied the forms of its primitive factors. Later, Lehmer [3] made a detailed study of this sequence of numbers. The theory of Zp -extensions is one of the most fruitful areas of research in number theory. A beautiful result in this area is the theorem of Iwasawa which describes the behavior of the p-part of the class number in a Zp -extension of number fields. Iwasawa Theorem ([7], Theorem 13.13.) — Let K∞ /K be a Zp -extension and K∞ = with [Kn : K] = pn . Let pen S+∞ n=0 Kn be the exact power of p dividing the class number of Kn . Then there exist integers λ ≥ 0, µ ≥ 0 and ν, all independent of n, and an integer n0 such that en = λn + µpn + ν, f or all n ≥ n0 . By the structure of Λ-modules, one sees that pen is indeed the value of the characteristic polynomial of some Λ-module at special points. From this viewpoint, in [2], the authors prove an analogue of Iwasawa’s theorem for higher K-groups of curves over finite fields. Let X be a smooth projective curve of genus g over a finite field F with q elements. For m ≥ 1, let Xm be the curve X over the finite field Fm , the m-th extension of F. For 1 ≤ i ≤ 2g, denote by πi the characteristic roots of the Frobenius endomorphism φ. Set 2g Y gn,m (x1 , x2 , · · · , x2g ) = (1 − (q n xi )m ). i=1 We have ]K2n (Xm ) = gn,m (π1 , π2 , · · · , π2g ), where K2n (Xm ) is the K-group of the smooth projective curve Xm . Let p be a prime. Denote the p-primary part of the order of K2n (Xpt ) by pen,p (t) , i.e., en,p (t) = ordp (]K2n (Xpt )). Theorem 1.1 ([2]) — There exist integers λn,p ≥ 0, νn,p and a positive integer Tn,p such that en,p (t) = λn,p t + νn,p , f or all t ≥ Tn,p . Let f (x) = x2 − P x − Q ∈ Z[x] and g = {gn (x1 , x2 )}n∈N , where g0 (x1 , x2 ) = 0, gn (x1 , x2 ) Pn−1 k n−1−k = , n ≥ 1. It is well-known that the recurring series ∆n (f, g) is the Lucas sek=0 x1 x2 quences Ln with parameters P and Q. The following results are consequences of the well-known properties of the Primitive Divisor Theorem for Lucas sequences. NUMERICAL FACTORS OF ∆n (f, g) 703 Theorem 1.2 — ([1]) Let n ∈ N∗ and p a prime. (1) If p - Ln Lp , then ordp (Lnpt ) = 0, for all t ∈ N. (2) If p|Ln Lp , then there exist integers νn,p and Tn,p such that ordp (Lnpt ) = t + νn,p , f or all t ≥ Tn,p . (3) Let n ∈ N∗ and p, q be two different primes. Then there exists a positive integer Tn,p,q such that ordq (Lnpt ) = ordq (LnpTn,p,q ), for all t ≥ Tn,p,q , i.e., the numbers ordq (Lnpt ) are stable when t is sufficiently large. (4) Let Sn,p (t) be the set of all primes which divide Lnpt . Then ]Sn,p (t) −→ +∞ as t −→ +∞. In this paper, we generalize Pierce’s recurring series {∆n }n∈N defined by (3) and {gn,m (π1 , π2 , · · · , π2g )}m∈N defined above to the recurring series {∆n,m }n∈N for any integer Q m ∈ N∗ , where ∆n,m = ri=1 (mn − αin ). We give a detailed study of essential and characteristic factors of ∆n,m especially as regards sequences of numbers. Let p be a prime and n, m ∈ N∗ , we establish an analogue of Iwasawa’s theorem for the orders ordp (∆npt ,m ) as follows. Theorem 3.7 — Let p be a prime. Fix integers n, m ∈ N∗ , let pen,m,p (t) be the p-primary part of ∆npt ,m for t ∈ N. (1) If p - ∆n,m , then en,m,p (t) = 0 for all t ∈ N. (2) If p|∆n,m , then there exist integers λn,m,p ≥ 1 and νn,m,p , both independent of t, and an integer Tn,m,p such that en,m,p (t) = λn,m,p t + νn,m,p , f or all t ≥ Tn,m,p . On the other hand, let p, q be two different primes, we prove that the numbers ordq (∆npt ,m ) are stable when t is sufficiently large (See Theorem 3.9). We also prove that the number of prime factors of ∆npt ,m goes to infinity as t goes to infinity. (See Corollary 3.10). 2. FACTORIZATION OF ∆n,m (m) Let the notation be as in §1. In this section, fix an integer m ∈ N∗ , define gm = {gn (X)}n∈N as follows gn(m) (x1 , x2 , . . . , xr ) = r Y (mn − xni ). i=1 704 QINGZHONG JI AND HOURONG QIN 2.1 Definition of ∆n,m — Let α1 , . . . , αr be the roots of the polynomial f (x) ∈ Z[x] defined by (1). Then ∆n,m is defined by ∆n,m = ∆n (f, gm ) = gn(m) (α1 , α2 , . . . , αr ) = r Y (mn − αin ). (4) i=1 Pierce [4] and Lehmer [3] listed many properties of ∆n = ∆n,1 . In this section, we will generalize all results in [3] concerning ∆n to the case ∆n,m , for all n, m ∈ N∗ . We would like point out that the idea used here is similar to that in [3]. Remark 2.2 : (1) The polynomial f (x) can be viewed as a characteristic polynomial of some r × r matrix A, for example, A= 0 0 ··· 0 −ar 1 0 · · · 0 −ar−1 0 1 · · · 0 −ar−2 . .. .. .. .. . . . . 0 0 · · · 1 −a1 Then f (x) = |xE − A| and ∆n,m = |mn E − An |, (5) where |B| = det(B) for any square matrix B. (2) Let α be a root of f (x). If f (x) is irreducible, then ∆n,m = NK/Q (mn − αn ), where K = Q(α) and NK/Q is the norm map from the field K to Q. (3) Since (m, ar )|∆n,m , for our purposes, we always assume (m, ar ) = 1 in this section. 2.3 Essential and characteristic factors of ∆n,m Let Φδ (x, y) be the δth homogeneous cyclotomic polynomial, i.e., Φδ (x, y) = δ Y (x − yζδi ) (6) i=1 (i,δ)=1 where ζδ is a primitive δth root of unity. Then we define the integer Φ∗δ,m by Φ∗δ,m = r Y i=1 Φδ (m, αi ). (7) NUMERICAL FACTORS OF ∆n (f, g) It follows from the formula xn − y n = Q δ|n Φδ (x, y) ∆n,m = Y 705 that Φ∗δ,m . (8) δ|n This gives a partial factorization of ∆n,m into integer factors. If we assume that each ∆, whose first subscript is a proper divisor of n, has been factored, the complete factorization of ∆n,m depends only on that of Φ∗n,m . For this reason we call this latter number the essential factor of ∆n,m . On the other hand, we may consider the prime factors of ∆n,m . Similarly, the prime factors of ∆n,m which do not divide ∆d,m , where d is a proper divisor of n, are called the characteristic prime factors of ∆n,m . The concepts of essential factor and characteristic prime factors were introduced by Lehmer [3]. Lemma 2.4 — The essential factor Φ∗n,m of ∆n,m contains all the characteristic prime factors of ∆n,m . P ROOF : By (8) a characteristic prime factor p of ∆n,m must divide Φ∗δ,m for some divisor δ of n. If δ were less than n, and hence p would divide ∆δ,m , contrary to the definition of p. Therefore δ = n 2 and the lemma follows. Lemma 2.5 — A characteristic prime factor p of ∆n,m cannot divide n. P ROOF : If possible, let n = pδ. Suppose f (x) = |xE − A| for some matrix A. Then by the multinomial theorem modulo p and (5), we have 0 ≡ ∆n,m ≡ ∆pδ,m ≡ |mpδ E − Apδ | ≡ |mδ E − Aδ |p ≡ |mδ E − Aδ | ≡ ∆δ,m (mod p). This contradicts the hypothesis that p is a characteristic factor of ∆n,m . 2 Remark : (1) It is not true that the essential factor of ∆n,m is made up exclusively of characteristic prime factors (See [3], p. 462). (2) The essential factor Φ∗n,m may, however, have a factor in common with n. (3) If f is reducible over the rational field so that f = f1 f2 , then ∆n,m (f ) = ∆n,m (f1 )∆n,m (f2 ) 706 QINGZHONG JI AND HOURONG QIN is a factorization into integers. Hence for our purposes we may suppose that f is irreducible. The following result is a generalization of [7], Lemma 2.9. Theorem 2.6 — If pe (e > 0) is the highest power of a characteristic prime factor p of ∆n,m (f ), where f is irreducible and of degree r, and if w is the order of p mod n, then w ≤ r and e is divisible by w. 2 P ROOF : It is similar to the proof of [3], Theorem 3. 2.7 The recurring series for ∆n,m In order to render the factorization of ∆n,m practical, it is first necessary to have a simple method of calculating its actual value. This is done with the help of a polynomial Mm (x) uniquely determined by f (x) and m in the following manner. Let f0 (x) = x − mr , Qr (x − mr−1 αi ), f1 (x) = i=1 Q f2 (x) = 1≤j<i≤r (x − mr−2 αi αj ), ··· ··························· fr (x) = x − α1 α2 · · · αr = x − (−1)r ar . Mm (x) is defined as the least common multiple of these f 0 s. That is, Mm (x) = [f0 (x), f1 (x), . . . , fr (x)] = xq + A1 xq−1 + A2 xq−2 + · · · + Aq . Definition 2.8 — Let {xn }∞ n=0 be a recurring series such that xn+e + a1 xn+e−1 + · · · + ae−1 xn+1 + ae xn = 0, for all n ≥ 0. Then the polynomial xe + a1 xe−1 + · · · + ae−1 x + ae is called the scale of the recurring series {xn }∞ n=0 . Theorem 2.9 — The numbers ∆0,m , ∆1,m , ∆2,m , ∆3,m , · · · form a recurring series whose scale is Mm (x). That is, for every n ≥ 0, ∆n+q,m + A1 ∆n+q−1,m + · · · + Aq−1 ∆n+1,m + Aq ∆n,m = 0. P ROOF : By the definition (4) of ∆n,m , we have ∆n,m r r X Y n n (m − αi ) = = i=1 X k=0 1≤i1 <···<ik ≤r (−1)k m(r−k)n αin1 · · · αink . (9) NUMERICAL FACTORS OF ∆n (f, g) 707 Hence we obtain q X Ai ∆n+q−i,m = i=0 r X X (−1)k m(r−k)n αin1 · · · αink Mm (m(r−k) αi1 · · · αik ) = 0, k=0 1≤i1 <···<ik ≤r where A0 = 1. This completes the theorem. 2 2.10 q-periodic ∆’s Lehmer [3] has proved that ∆n,1 is a periodic function of proper period τ if and only if f (x) = Φτ (x, 1). In this subsection, for a fixed integer m, we will consider the periodic properties of ∆n,m . Definition 2.11 — Suppose F : Z −→ C is a number theory function. We call F a q-periodic function of period τ, if there exists a function λ(n) such that F (qτ + k) = λ(q)F (k), for all q, k ∈ Z. (10) The function λ is called a periodic factor of F. We also call F q-periodic with respect to λ. A positive integer τ is called a proper period of a q-periodic function F, if for any positive integer T < τ, F is not q-periodic of period T. Remark 2.12 : (1) It is obvious that a periodic function F is q-periodic, in this case λ(n) = 1, for all n ∈ Z. (2) If the function F is defined over N and τ is a positive integer such that F (qτ + k) = λ(q)F (k), for all q, k ∈ N, for some function λ(n) defined over N, then F can be extended to a q-periodic function defined over Z. In this case we also call F a q-periodic function defined over N. Lemma 2.13 — If F 6= 0 is a q-periodic function with respect to a function λ, then λ(n) 6= 0 for all n ∈ Z and λ : Z −→ C∗ is a group homomorphism. P ROOF : Suppose there exists an integer n0 ∈ Z such that λ(n0 ) = 0. By (10), for any n ∈ Z, we have F (n) = F (n0 T + (n − n0 T )) = λ(n0 )F (n − n0 T ) = 0. This contradicts the assumption F 6= 0. It is easy to see that λ(0) = 1 and λ(m+n) = λ(m)λ(n) for all m, n ∈ Z. Hence λ is a group homomorphism. 2 Lemma 2.14 — Suppose τ is a proper period of a q-periodic function F. If T is a period of F, then τ |T. 708 QINGZHONG JI AND HOURONG QIN P ROOF : First we prove that q-periodic functions have properties similar to those of periodic functions. Let T > 0 be a period of a q-periodic function F (n), i.e., there exists a function λ(n) such that F (qT + k) = λ(q)F (k), for all q, k ∈ Z. Then we have (i) for any a ∈ Z, aT is a period of F. In fact, set λa (n) = λ(an), then F (qaT + k) = λ(aq)F (k) = λa (q)F (k), for all q, k ∈ Z. (ii) if T1 and T2 are two periods of F, then T1 + T2 is also a period of F. Assume λi is corresponding to Ti , i = 1, 2. Set λ(n) = λ1 (n)λ2 (n), then F (q(T1 + T2 ) + k) = λ1 (q)F (qT2 + k) = λ1 (q)λ2 (q)F (k) = λ(q)F (k), for all q, k ∈ Z. Suppose τ - T. Then T = q0 τ + b where q0 , b ∈ Z and 0 < b < τ. By (i) and (ii) above, we obtain that b = T − q0 τ is a period of F. This contradicts the fact that τ is a proper period of F. 2 For a fixed integer m, it may happen that ∆n,m is a λ-periodic function of n. In this case we have Theorem 2.15 — A necessary and sufficient condition for ∆n,m to be q-periodic function of n of proper period τ is that f (x) = Φτ (x, m), where Φτ (x, y) is the τ -th homogeneous cyclotomic polynomial defined by (6). P ROOF : If ∆n,m is q-periodic of proper period τ , then ∆τ,m = 0. Hence f has a root α for which ατ = mτ . Then there exists a primitive kth root ζk of unity such that α = mζk . Since f is irreducible all its roots are mζki , 1 ≤ i ≤ k, (k, i) = 1, so that f (x) = Φk (x, m), where k is some divisor of τ. But ∆n,m is of period k, for if n, j are any integers ≥ 0, ∆nk+j,m = Y Y (mnk+j − αink+j ) = mϕ(k)kn (mj − αij ) = λ(n)∆j,m , i i where λ(n) = mϕ(k)kn is the periodic factor. Hence by Lemma 2.14, τ is a divisor of k. Therefore τ = k and f (x) = Φτ (x, m). 2 3. I WASAWA T HEORY OF ∆n,m Let the notation be as in §2. For our purposes, in this section, we make the following hypothesis: (H 1) f (x) is defined by (1) and irreducible. (H 2) Fix an integer m satisfying (m, ar ) = 1. NUMERICAL FACTORS OF ∆n (f, g) 709 (H 3) f (x) 6= ΦT (x, m) for all T ∈ N∗ . Let K = Q(α1 , α2 , . . . , αr ) be the splitting field of f (x) over the rational number field Q, OK the ring of algebraic integers of K. For any prime p, let P be a prime ideal of K lying above p. Theorem 3.1 — Let n ∈ N and p be a prime factor of ∆n,m . Then, for any positive integer t ∆ . satisfying p|t, we have p| ∆nt,m n,m P ROOF : By the formula (4), the condition p|∆n,m implies mn ≡ αin (mod P) for some i(1 ≤ i ≤ r). If p|t, then we have r r X t−1 Y Y mnt − αjnt ∆nt,m n(t−1−k) n(t−1) = ≡ tm mnk αj ≡ 0 (mod P). ∆n,m mn − αjn j=1 j=1 j6=i k=0 ∆ Hence p| ∆nt,m . n,m 2 Corollary 3.2 — (1) Let n ∈ N∗ and p a prime factor of ∆n,m . Then, for all t ∈ N, we have pe+t |∆npt ,m , where e = ordp (∆n,m ). (2) Let n, t ∈ N∗ . Then we have (∆n,m )t |∆n(∆n,m )t−1 ,m . P ROOF : (1) It follows easily by induction on t. (2) It follows trivially from (1) and the fact: ∆n1 ,m |∆n2 ,m , if n1 |n2 . 2 Theorem 3.1 is about divisibility. The next result will be about non-divisibility. First, a definition. If p is a prime, put d(p) = lcm1≤i≤[K:Q] {pi − 1}. 2 Theorem 3.3 — Let n, t ∈ N∗ . Suppose p is a prime such that p - ∆n,m and (t, d(p)) = 1. Then (i) p - ∆nt,m ; (ii) p - ∆npx ,m for any x ∈ N. P ROOF : It is clear that (ii) follows (i). Hence it suffices to prove (i). If p|∆nt,m , then mnt ≡ αint (mod P) for some i (1 ≤ i ≤ r). (11) If αi ≡ 0 (mod P), then m ≡ 0 (mod P). Hence p|(ar , m), this contradicts the assumption (ar , m) = 1. So m, αi 6∈ P. By (11), we have µ n ¶t αi ≡1 mn αn (mod P). αn But ( min )d(p) ≡ 1 (mod P) and (t, d(p)) = 1, we have min ≡ 1 (mod P), i.e., mn ≡ αin Q (mod P). Hence ∆n,m = rj=1 (mn − αjn ) ≡ 0 (mod P) contradicts p - ∆n,m . Hence p - ∆nt,m . This completes the proof. 2 710 QINGZHONG JI AND HOURONG QIN Lemma 3.4 — Let n be the smallest integer such that p|∆n,m . Then n|d(p). P ROOF : From the proof of Theorem 3.3, if p|∆n,m , then there exists an index i (1 ≤ i ≤ r) such that mn ≡ αin (mod P) and m, αi 6∈ P. (i) Assume m ≡ αi (mod P). Then ∆1,m = (ii) Assume m 6≡ αi (mod P). Then ( αi m Qr j=1 (m − αj ) ≡ 0 (mod P), so n = 1. 6≡ 1 (mod P) and αi n ) ≡ 1 (mod P). m From the definition of n, it follows that n is the order of ( αmi )d(p) αi m (mod P). On the other hand, ≡ 1 (mod P). Hence n|d(p) as asserted. 2 Corollary 3.5 — Let p be a prime. Then p|∆1,m if and only if m ≡ αi (mod P) for some i (1 ≤ i ≤ r). Let Qp be the p-adic completion of Q. Let Q and Qp be the algebraic closures of Q and Qp , respectively. Let ρ be an embedding of Q into Qp . We simply rename ρ(a) as a. S We will keep the notation ordp for the additive valuation from Qp to Q {∞}, extended by the S standard additive valuation ordp from Qp to Z {∞}, namely, if α ∈ Qp , then ordp (α) = [Qp (α) : Qp ]−1 ordp (NQp (α)/Qp (α)). Here NQp (α)/Qp is the usual norm map from Qp (α) to Qp . Lemma 3.6 ([6], p. 172-174) — Let p and q be different primes. For n ≥ 1, let ξ ∈ Qp be any primitive pn -th root of unity. Then the following results hold. (1) ordp (ξ − 1) = 1 pn−1 (p−1) and ordq (ξ − 1) = 0. (2) Let α ∈ Qp be integral over Zp . t (i) If ordp (α − 1) = 0, then ordp (αp − 1) = 0 for all positive integers t ≥ 1. (ii) If ordp (α − 1) > 0, then there exist an integer t0 and a constant c depending on α such that t ordp (αp − 1) = t + c, for all t ≥ t0 . In fact, t0 and c can be chosen as t0 = min{t ∈ Z| 1 pt−1 (p − 1) < ordp (α − 1)}, NUMERICAL FACTORS OF ∆n (f, g) and c = ordp (α − 1) + X 711 [ordp (α − ξ) − ordp (1 − ξ)], 16=ξ∈S where S is the set of pi -th roots of unity, 1 ≤ i < t0 . (3) Let β ∈ Qq be integral over Zq . t (i) If ordq (β − 1) > 0, then ordq (β p − 1) = ordq (β − 1) > 0, for all t ≥ 1. (ii) If ordq (β − 1) = 0, then there exists an integer t0 ≥ 0 such that, for all t ≥ t0 , t t0 ordq (β p − 1) = ordq (β p − 1). Let n ∈ N∗ and p a prime. Set en,m,p (t) = ordp (∆npt ,m ) for t ∈ N. Theorem 3.7 — Let n ∈ N∗ and p a prime. (1) If p - ∆n,m , then en,m,p (t) = 0 for all t ∈ N. (2) If p|∆n,m , then there exist integers λn,m,p ≥ 1, νn,m,p and Tn,m,p such that en,m,p (t) = λn,m,p t + νn,m,p , f or all t ≥ Tn,m,p . P ROOF : By Theorem 3.3, if p - ∆n,m , then en,m,p (t) = 0, for all t ∈ N. Hence, it suffices to prove (2). Assume p|∆n,m . Without loss of generality, we may assume n is the smallest integer such that p|∆n,m . By Lemma 3.4, n|d(p), hence (p, n) = 1. Let ζn be a primitive nth root of unity. Then ∆n,m = r r n−1 r n−1 Y Y Y Y Y αi (mn − αin ) = (m − αi ζnj ) = mrn (1 − ζnj ). m i=1 i=1 j=0 i=1 j=0 Note that p|∆n,m implies ordp (m) = 0 (see the proof of Theorem 3.3). Hence we have ordp (1 − αi j m ζn ) ≥ 0 for all 1 ≤ i ≤ r, 0 ≤ j ≤ n − 1. For each i (1 ≤ i ≤ r), we claim αi j0 m ζn ) > 0. In fact, if there 0 and ordp (1− αmi ζnj2 ) > 0, then ordp (αi ) = 0 that there is at most one index j0 (0 ≤ j0 ≤ n − 1) such that ordp (1 − exist 0 ≤ j1 < j2 ≤ n−1 such that ordp (1− αmi ζnj1 ) > and ordp (1 − ζnj2 −j1 ) = ordp ( αi αi αi j 1 ζn (1 − ζnj2 −j1 )) = ordp ((1 − ζnj2 ) − (1 − ζnj1 )) > 0. m m m This contradicts (p, n) = 1. Set λn,m,p = ]{i | 1 ≤ i ≤ r, there exists an index j such that ordp (1 − αi j ζ ) > 0}. m n 712 QINGZHONG JI AND HOURONG QIN The condition p|∆n,m implies λn,m,p ≥ 1. On the other hand, we have ∆npt ,m = = = = Qr t npt − αnp ) i=1 (m i Qr Qn−1 pt pt j i=1 j=0 (m − αi ζn ) Q t Q αi p t j mrnp ri=1 n−1 j=0 (1 − ( m ) ζn ) Qn−1 t Q t mrnp ri=1 j=0 (1 − ( αmi ζnj )p ) (12) since (n, p) = 1. By (2) of Lemma 3.6, there exist integers νn,m,p and Tn,m,p such that for all t ≥ Tn,m,p , we have en,m,p (t) = ordp (∆npt ,m ) P Pn−1 t = ri=1 j=0 ordp (1 − ( αmi ζnj )p ) = λn,m,p t + νn,m,p , where the integers λn,m,p and νn,m,p are independent of t. 2 Q Remark 3.8 : For each n ∈ N, set fn (x) = ri=1 (x − αin ). Let p be a prime factor of ∆n,m . Factor fn (x) over Fp [x] as follows: fn (x) = p1 (x)e1 p2 (x)e2 · · · ps (x)es where p1 (x), p2 (x), . . . , ps (x) ∈ Fp [x] are non-associate irreducible polynomials with multiplicity ei ≥ 1 (1 ≤ i ≤ s). If e1 = e2 = · · · = es = 1, then λn,m,p = 1, i.e., there exists a unique index i0 such that ordp (mn − αin0 ) > 0. In fact, if there exist 1 ≤ i < j ≤ r such that ordp (mn − αin ) > 0 and ordp (mn − αjn ) > 0, then ordp (m) = 0 and so ordp (αin − αjn ) = ordp ((mn − αjn ) − (mn − αin )) > 0. Hence αin is a root of fn (x) with multiplicity at least 2 over Fp which is the algebraic closure of Fp . This contradicts the assumptions e1 = e2 = · · · = es = 1. Theorem 3.9 — Let n ∈ N∗ and p, q be two different primes. Then there exists a positive integer Tn,m,p,q such that ordq (∆npt ,m ) = ordq (∆npTn,m,p,q ,m ), f or all t ≥ Tn,m,p,q , i.e., the numbers ordq (∆npt ,m ) are stable when t is sufficiently large. P ROOF : Without loss of generality, we may assume (n, p) = 1 and ordq (m) = 0. On the other hand, by (12), we have rnpt ∆npt ,m = m r n−1 Y Y αi t (1 − ( ζnj )p ). m i=1 j=0 NUMERICAL FACTORS OF ∆n (f, g) 713 (i) (i) For each i (1 ≤ i ≤ r), we divide the set {j|0 ≤ j ≤ n − 1} = S1 ∪ S2 in such a way (i) (i) t pj (i) that for each j ∈ S1 , there is a tj ≥ 0 such that ordq (1 − ( αmi ζnj ) (i) ) > 0, and for j ∈ S2 , the (i) equality ordq (1 − ( αmi ζnj )p ) = 0 holds, for all t ≥ 0. Set Tn,m,p,q = max 1≤i≤r {tj }. Then, for all t (i) j∈S1 t ≥ Tn,m,p,q , by Lemma 3.6, we have ordq (∆npt ,m ) = = Pr Pn−1 Pr P i=1 i=1 j=0 ordq (1 − ( αmi ζnj )p ) t (i) j∈S1 ordq (1 − (i) t j ( αmi ζnj )p ). Since the last sum does not depend on t, the result follows. 2 Corollary 3.10 — Let Sn,m,p (t) be the set of all primes which divide ∆npt ,m . Then ]Sn,m,p (t) −→ +∞ as t −→ +∞. P ROOF : Suppose that there exists integer t0 such that for all t ≥ t0 , Sn,m,p (t) = Sn,m,p (t0 ). By Theorem 3.7 and Theorem 3.9, it would follow that ∆npt ,m would be equal to a constant times pλn,m,p t for large t, i.e., there exist positive constant numbers T and c such that |∆npt ,m | = cpλn,m,p t (13) for all t ≥ T. On the other hand, the assumption (H3) implies that m 6= |αi |, 1 ≤ i ≤ r. Set S1 = {i | m > |αi |}, S2 = {i | m < |αi |}, b = Y max{|m|n , |αi |n }. 1≤i≤r If m = 1, then S2 6= ∅. Hence, for all m ≥ 1, we have b > 1 and lim t−→+∞ |∆npt ,m | bpt Q = = = t−→+∞ t 1≤i≤r lim bpt t t |mnp −αnp | Q |mnp −αnp | i i · i∈S2 t i∈S np 1 m |αi |npt t−→+∞ Q Q t t lim |1 − ( αmi )np | · i∈S2 |1 − ( αmi )np | i∈S 1 t−→+∞ lim Q t |mnp −αnp | i t t = 1. t Therefore for sufficiently large t, we have |∆npt ,m | > abp for some constant a > 0. Clearly, this is incompatible with (13) just given. 2 At last, we give the following definition. Definition 3.11 — A sequence of integers {an } is called an Iwasawa sequence if for any positive integer m and prime p, there exist integers λ, T ∈ N and ν ∈ Z such that ordp (ampt ) = λt + ν, f or all t ≥ T. 714 QINGZHONG JI AND HOURONG QIN Example 3.12 : Let m be any positive integer. Then the sequence of binomial coefficients {Cnm }n≥m is an Iwasawa sequence. In fact, by Kummer Theorem, m ordp (Cnp t ) = t + ν, f or all t ≥ T, where T = max{0, [logp m] − ordp (n) + 1} and ν = ordp (n) − ordp (m). ACKNOWLEDGEMENT We would like to thank the referee for reading the manuscript carefully and providing valuable comments and suggestions. R EFERENCES 1. Yu. F. Bilu, G. Hanrot and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers, J. reine angew Math., 539 (2001), 75-122. 2. Q. Ji and H. Qin, Higher K-groups of smooth projective curves over finite fields, Finite Fields and Their Applications, 18 (2012), 645-660. 3. D. H. Lehmer, Factorization of certain cyclotomic funtions, Annals of Math., 34(2) (1933), 461-479. Qn 4. T. A. Pierce, The numerical factors of the arithmetic forms i=1 (1 ± αim ), Annals of Math., 18 (1916), 53-64. 5. P. Ribenboim, My Numbers, My Friends, Springer-Verlag 2000. 6. M. Rosen, Number Theory in Function Fields, GTM 210, Springer-Verlag, New York, 2002. 7. L. C. Washington, Introduction to cyclotomic fields (Second Edition), GTM 83, Springer-Verlag, New York, 1996.