FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER

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1 #A30 INTEGERS 10 (010), FLAT CYCLOTOMIC POLYNOMIALS OF ORDER FOUR AND HIGHER Nahan Kaplan Deparmen of Mahemaic, Harvard Univeriy, Cambridge, MA Received: 7/15/09, Revied: 3/1/10, Acceped: 3/18/10, Publihed: 7/19/10 Abrac In hi aricle we prove a reul abou e of coefficien of cycloomic polynomial. We hen give corollarie relaed o fla cycloomic polynomial and eablih he fir known infinie family of fla cycloomic polynomial of order four. We end wih ome queion relaed o fla cycloomic polynomial of order four and five. 1. Inroducion The nh cycloomic polynomial i he monic polynomial whoe roo are he primiive nh roo of uniy. I i defined by n a=1 (a,n)=1 (x e πia/n ). The degree of Φ n i φ(n), where φ i he Euler oien funcion. We ay ha he order of a cycloomic polynomial i he number of odd prime dividing n. We can facor x n 1 = d n Φ d (x). The following propoiion allow u o focu on odd quarefree value of n for he re of he paper. See [8] for a proof of hi and for oher general reul abou cycloomic polynomial. Propoiion 1. Le p be a prime. If p n hen Φ pn (x) = Φ n (x p ). If p n hen Φ pn (x) = Φ n (x p )/Φ n (x). If n > 1 i odd hen Φ n ( x).

2 INTEGERS: 10 (010) 358 Le φ(n) k=0 a n(k)x k. We pu a n (k) = 0 if k < 0 or k > φ(n). Le V n = {a n (k) : 0 k φ(n)} denoe he e of coefficien of Φ n (x). I i eay o verify from he definiion of Φ n (x) ha for n > 1, x φ(n) Φ n (x 1 ). Thi implie ha for n > 1, V n = {a n (k) : 0 k φ(n) }. We ay ha A(n) = max k { a n(k) } i he heigh of Φ n (x). Several recen paper have udied n for which A(n) i large, for example, [1, 6]. I i alo inereing o aemp o claify n uch ha A(n) i mall. If A(n) = 1 we ay ha Φ n (x) i fla. I i eay o how ha, for odd prime p < q, we have V (p) = {1} and V (pq) = { 1, 0, 1} and herefore A(p) = A(pq) = 1. Bachman gave he fir infinie family of fla cycloomic polynomial of order hree [3], and hi family wa expanded by Kaplan [7], who proved he following. Theorem. ([7]) Le p < q < r be prime uch ha r ±1 (mod pq). Then A(pqr) = 1. There exi fla cycloomic polynomial of order hree ha are no of hi form. I would be an inereing and difficul problem o claify hem. Beier ha claified all fla cycloomic polynomial of he form Φ 3qr (x) [4], bu no much i known abou fla cycloomic polynomial of he form Φ pqr (x) for p 5 or fla cycloomic polynomial of order greaer han hree. Recenly, Broadhur [5] ha made ome conjecure abou fla cycloomic polynomial of order hree. Le p < q < r be odd prime wih w he unique ineger 0 < w pq 1 aifying r ±w (mod pq). (i) If w = 1 hen we ay ha [p, q, r] i of Type 1. (ii) If w > 1, q 1 (mod pw), and p 1 (mod w) hen we ay ha [p, q, r] i of Type. (iii) If w > p, q > p(p 1), q ±1 (mod p) and w ±1 (mod p), and in he cae where w 1 (mod p) we have wp q + 1 and wp q 1, hen we ay ha [p, q, r] i of Type 3. Conjecure. ([5]) (i) If [p, q, r] i of Type 1 or, hen A(pqr) = 1. (ii) If [p, q, r] i no of Type 1,, or 3, hen A(pqr) > 1. (iii) If [p, q, r] i of Type 3, hen A(pqr) = 1 if and only if Φpq(x ) Φ pq(x) i fla for he malle poiive ineger uch ha 1 (mod p) and ±r (mod pq). Noe ha Theorem ae ha if [p, q, r] i of Type 1, hen A(pqr) = 1. Thi conjecure, if rue, goe a long way oward a complee claificaion of fla cycloomic polynomial of order hree. I would remain o give condiion on [p, q, r] of

3 INTEGERS: 10 (010) 359 Type 3 for which Φpq(x ) Φ pq(x) i fla for he decribed in he conjecure. Broadhur ha alo conjecured bound on he number of [p, q, r] of Type 3 which give fla cycloomic polynomial [5]. The main reul of hi paper, Theorem 4, i a naural generalizaion of Theorem in [7]. Theorem 3. (Kaplan, 007) Le p < q < r < be prime uch ha r ± (mod pq). Then A(pqr) = A(pq). Le Ψ n (x) = xn n φ(n) 1 c k x k denoe he nh invere cycloomic polynomial. We can eaily ee ha deg(ψ n (x)) = n φ(n). We pu c k = 0 if k < 0 or k > n φ(n). Thee polynomial have been udied recenly by Moree [10]. They will be ued in he proof of Theorem 4. k=0. The Main Reul In hi paper we will prove he following reul which applie o cycloomic polynomial of arbirary order, bu require lighly ronger aumpion han Theorem 3. Theorem 4. Le < p 1 < p < < p r be prime and n = p 1 p r. Le, be prime aifying n < < and (mod n). Then V n = V n. Proof. We may uppoe ha r and herefore n 15 ince for any odd prime p < q we have V (pq) = { 1, 0, 1}. For impliciy we will change our noaion lighly. Le and (p 1 1) (p r 1)( 1) i=0 (p 1 1) (p r 1)( 1) i=0 b i x i, d i x i. We will fir how ha V n V n by howing ha for any coefficien b l V n, here i a coefficien d m V n wih d m = b l.

4 INTEGERS: 10 (010) 360 We have Φ n(x ) ( x n 1 Φ n(x) ) Φ n (x ) x n 1 = Ψ n(x)φ n (x ) x n. 1 Noe ha deg(ψ n (x)) = n φ(n) = n (p 1 1) (p r 1). We have aumed ha > deg(ψ n (x)). Similarly, we have Φ n(x ) Ψ n(x)φ n (x ) x n. 1 By expanding 1 x n 1 = (1 + xn + x n + ), we have Ψ n (x)φ n (x )(1 + x n + x n + ), and Ψ n (x)φ n (x )(1 + x n + x n + ). Le (p 1 1) (p r 1) j=0 a j x j, and Ψ n (x) = n φ(n) i=0 c i x i. The erm of Ψ n (x)φ n (x ) are of he form c i a j x i+j. Similarly he erm of Ψ n (x)φ n (x ) are of he form c i a j x i+j. Since (mod n), i + j i + j (mod n). For a fixed l, conider he e of (i, j) uch ha c i 0 and i + j = l. Since c i 0 implie ha 0 i n φ(n) <, here i a mo one pair (i, j) in hi e. Similarly for a fixed m, he e of (i, j) uch ha c i 0 and i + j = m ha a mo one elemen. We ee ha b l = (i,j) c i a j, where he um i aken over all pair (i, j) uch ha i + j l, i + j l (mod n), and c i 0. Similarly, d m = (i,j) c i a j, where he um i aken over all pair (i, j) uch ha i+j m, i+j m (mod n), and c i 0.

5 INTEGERS: 10 (010) 361 For any ineger l wih 0 l deg(φ n (x)) = φ(n)( 1), we can wrie l = k+α where k, α Z and 0 α <, in a unique way. Noe ha k < φ(n). Now le m = k + α. Since k + α φ(n)( 1), we have k + α φ(n)( 1) + k( ) < φ(n)( 1) + φ(n)( ) = deg(φ n (x)). Suppoe c i 0. We have i + j k + α if and only if j k + α i alway an ineger we have i+j k+α if and only if j k + α i α i for α < i.. Since j i. If α i, hen = 1 = 0. Since α 0 and c i 0 implie i n φ(n) <, we have α i Similarly i + j k + α if and only if j k + α i. Since < < i α i α < < we ee ha α i = α i. Therefore i + j k + α if and only if i + j k + α, and b l = d m. So for any coefficien b l of Φ n (x), here i a coefficien d m of Φ n (x) wih d m = b l and V n V n. Now we will how ha V n V n by howing ha for any coefficien d m V n here i a coefficien b l V n uch ha b l = d m. If m deg(φn(x)) hen m = deg(φ n (x)) m deg(φn(x)). Since d m = d m, wihou lo of generaliy we may uppoe ha m deg(φn(x)). Given m we can wrie m = k + β where k, β Z and 0 β < in a unique way. Noe ha k < φ(n). Suppoe c i 0. A in he previou paragraph we have i + j k + β if and only if j k + β i. Le α β (mod n) wih 0 α < n <. Now conider l = k + α. We have k + α < ( φ(n) + 1) (φ(n) 1) < φ(n)( 1) = deg(φ n (x)) ince 4 φ(n) for all n 7. If β < i, hen β < n and o α = β. We ee ha β i = α i = 1 and b l = d m. Suppoe ha β i. Then β i = 0. Since α < n <, we have α i 0. If α i = 0, hen clearly i + j l if and only if i + j m, and b l = d m. Suppoe here exi a pair (i, j) uch ha i + j l (mod n), c i 0, i + j k + β, bu i + j > k + α. Thi implie ha j k and j > k + α i. Therefore α i = 1 and j = k. So i + k k + α (mod n) and i > α. Thi implie i α 0 (mod n). Since i α > 0 we have i n > n φ(n), which conradic c i 0. Thi implie ha uch a pair (i, j) doe no exi. So i + j k + β if and only if i + j k + α, and herefore b l = d m. So for any coefficien d m of Φ n (x), here i a coefficien b l of Φ n (x) wih b l = d m, and hu V n V n.

6 INTEGERS: 10 (010) Some Conequence and Open Queion Several corollarie follow direcly from Theorem 4. Corollary 5. Le < p 1 < p < < p r be prime and n = p 1 p r. Le, be prime aifying n < < and (mod n). We have A(n) = A(n). I i unclear how much we can weaken he aumpion in Theorem 4 ha n < <. The reul i no rue if we imply require ha p r < <. For example V ( ) V ( ). Corollary 6. Le n = p 1 p p r be a produc of diinc odd prime. If here exi a prime > n uch ha Φ n (x) i fla, hen here are infiniely many fla cycloomic polynomial of order r + 1. In paricular, A(n) = 1 whenever i a prime uch ha > n and (mod n). Thi corollary follow from Dirichle heorem for prime in arihmeic progreion. We noe ha A( ) = 1. Corollary 7. There are infiniely many fla cycloomic polynomial of order four. In paricular, given any prime congruen o 1 modulo 465, A( ) = 1. Recenly Arnold and Monagan have inroduced improved mehod for quickly compuing he heigh of cycloomic polynomial and have made much of heir daa available online [1, ]. In paricular, here are 1389 fla cycloomic polynomial of order four wih n < They are all of he form n = pqr where q 1 (mod p), r ±1 (mod pq) and ±1 (mod pqr). We upec ha all fla cycloomic polynomial of order four are of hi form. In our limied compuaion i appear ha all of hee polynomial are fla. We alo have reaon o believe he following. Conjecure. If A(n) > 1 hen for any prime p, A(pn) > 1. I i unknown wheher here are any fla cycloomic polynomial of order greaer han four. There are none of order five wih n < [1, ]. For prime (p, q, r,, ) aifying q 1 (mod p), r 1 (mod pq), 1 (mod pqr) and 1 (mod pqr), Φ pqr (x) i no necearily fla. Andrew Arnold recenly compued he heigh of a cycloomic polynomial aifying hee congruence condiion. For (p, q, r,, ) = (3, 5, 9, 609, 6989), A(pqr) = A( ) =. Many of he above obervaion are baed on compuaion done by Tiankai Liu [9].

7 INTEGERS: 10 (010) 363 Acknowledgmen. I would like o hank Joe Gallian for running he Univeriy of Minneoa Duluh ummer reearch program where I wa fir inroduced o hi opic. I would like o hank Tiankai Liu for performing calculaion which were very helpful for he la ecion of hi paper and Andrew Arnold for anwering ome compuaional queion. I would like o hank he referee for everal ueful commen and Sam Elder for helpful dicuion relaed o hi projec. Reference [1] A. Arnold, M. Monagan, Calculaing cycloomic polynomial of very large heigh, ubmied o Mah. Comp. [] A. Arnold, M. Monagan, Daa on he heigh and lengh of cycloomic polynomial, available: hp:// ada6/cycloomic/. [3] G. Bachman, Fla cycloomic polynomial of order hree, Bull. London Mah. Soc. 38 (006), [4] M. Beier, Coefficien of he cycloomic polynomial, F 3qr (x), Fibonacci Quar., 16 (1978), [5] D. Broadhur, Fla ernary cycloomic polynomial, hp://ech.group.yahoo.com/group/primenumber/meage/0305. [6] Y. Gallo, P. Moree, Ternary cycloomic polynomial having a large coefficien, J. Reine Angew. Mah. 63 (009), [7] N. Kaplan, Fla cycloomic polynomial of order hree, J. Number Theory 17 (007), [8] H.W. Lenra, Vanihing um of roo of uniy, in: Proceeding, Bicenennial Congre Wikundig Genoochap (Vrije Univ., Amerdam, 1978), Par II, 1979, pp [9] T. Liu, peronal communicaion. [10] P. Moree, Invere cycloomic polynomial, J. Number Theory 19 (009),