Math. J. Okayama Univ. 54 (2012), 1–32 SOME REMARKS ON LUCAS PSEUDOPRIMES
Noriyuki Suwa
Abstract. We present a way of viewing Lucas pseudoprimes, Euler- Lucas pseudoprimes and strong Lucas pseudoprimes in the context of group schemes. This enables us to treat the Lucas pseudoprimalities in parallel to establish pseudoprimes, Euler pseudoprimes and strong pseudoprimes.
Introduction
Let p be a prime > 2. Then, as is well known, we have the following assertions: (1)(Fermat) If a is an integer prime to p, then ap 1 1 mod p; − ≡ p−1 a (2)(Euler) If a is an integer prime to p, then a 2 mod p; ≡ p (3) Put p 1 = 2sm with (m, 2) = 1. If a is an integer prime to p, then − k either am 1 mod p or a2 m 1 mod p for some k Mathematics Subject Classification. Primary 11Y11; Secondary 14L15. Key words and phrases. primality test, group scheme. Partially supported by The Research on Security and Reliability in Electronic Society, Chuo University 21st Century COE Program. 1 2 NORIYUKI SUWA view these pseudoprimalities using the language of group schemes because the ideas then seem much clearer, to use a phrase at the beginning of [3, Ch.III, 6]: it is profitable to view this pseudoprime construct using the lan- guage of finite fields, not just to be fashionable, but because the ideas then seem less ad hoc. It would narrow a way to reject the language of group schemes though it is not so popular yet. We use here only elementary facts on affine group schemes, for example, the contents in the first two chapters of the introductory book [Waterhouse, 13]. The main result is stated as Theorem 3.4, and the following assertion is the key for our argument. Corollary 2.8. Let P , Q and D be non zero integers, and let n be an odd integer > 1 with (n,DQ) = 1 and P 2 4Q D mod n. Put − ≡ D + 2Q P ξ = , U(D)(Z/nZ). 2Q 2Q ∈ Then: n ε(n) (1) n is a Lucas pseudoprime with respect to (P,Q) if and only if ξ − = I in U(D)(Z/nZ); (2) n is an Euler-Lucas pseudoprime with respect to (P,Q) if and only if Q n−ε(n) Q n−ε(n) either = 1 and ξ 2 = I, or = 1 and ξ 2 = I in n n − − U(D)(Z/nZ); (3) n is a strong Lucas pseudoprime with respect to (P,Q) if and only if k ξm = I or ξ2 m = I for some k Section 6. For example, the following assertion would reveal a part of inter- esting relations among various pseudoprimalities, some of which Grantham investigated in [5]. Proposition 6.7. Let P , Q be integers = 0 with D = P 2 4Q not a square, and let n be an odd composite with6 (n,DQ) = 1. Assume− that n is a Frobenius pseudoprime with respect to (P,Q). Then n is an Euler pseudoprime to base Q if and only if n is an Euler-Lucas pseudoprime with respect to (P,Q). It would be worth while to verify in our context the probable prime tests proposed by Kida [7], to examine the deterministic prime tests proposed by Gurevich-Kunyavskii [6] and to analyze the Frobenius pseudoprimalities defined by Grantham [4], [5]. We refer to original papers [1], [2], [4], [5], [15] and monographs [3], [11] for further topics on the Lucas pseudoprimalities, for example, the distribution of Lucas pseudoprimes and the accuracy of probable prime tests related with the Lucas pseudoprimalities. Notation a For a positive odd integer n and an integer a prime to n, denotes n the Jacobi symbol. Gm,Z: the multiplicative group scheme over Z GD, U(D): defined in 2.1 B˜ℓpsp(n, D), B˜eℓpsp(n, D), B˜sℓpsp(n, D): defined in 3.3 Bpsp(n), Bepsp(n), Bspsp(n): defined in [9] and recalled in 3.7 For a group scheme G and a commutative ring R, G(R) denotes the group of R-valued points of G. In particular, R× = Gm(R) stands for the multiplicative group of invertible elements of R. 1. Recall: Lucas pseudoprimalities In the section, we fix non-zero integers P , Q with P 2 4Q = 0 and put D = P 2 4Q. − 6 − Definition 1.1. Let P , Q be integers = 0, and put D = P 2 4Q. We assume that D = 0. Let α, β denote the6 roots of the quadratic− equation 2 6 t Pt + Q = 0. We define sequences Un n 0 and Vn n 0 by − { } ≥ { } ≥ αn βn U = U (P,Q)= − ,V = V (P,Q)= αn + βn. n n α β n n − The sequence Un n 0, Vn n 0 are called Lucas sequences associated to (P,Q). { } ≥ { } ≥ The following assertions are verified immediately from the definition. 4 NORIYUKI SUWA (1) U = 0, U = 1, U PU + QU = 0, V = 2, V = P , V 0 1 n+2 − n+1 n 0 1 n+2 − PVn+1 + QVn = 0. (2) V 2 DU 2 = 4Qn. n − n (3) 2Vn+m = VnVm + DUnUm, 2Un+m = UnVm + VnUm. In particular, 1 (4) V = (V 2 + DU 2)= V 2 2Qn = DU 2 + 2Qn, U = U V . 2n 2 n n n − n 2n n n Moreover, the following assertion is concluded readily from the recurrence relations (1): (5) Let p be a prime, and assume that p Q. If p P , then p U and p V | | | n | n for any n > 1. On the other hand, if p ∤ P , then p ∤ Un and p ∤ Vn for any n 1. ≥ Lemma 1.2. Let n be an odd integer > 1 with (n, D) = 1. If there exists k > 1 such that U 0 mod n, then n is prime to Q. k ≡ Proof. Assume that (n,Q) > 1. Let p be a prime divisor of (n,Q). By 1.1(5), if p ∤ P , then p ∤ Uk for any k > 1, which is a contradiction to the assumption. On the other hand, if p P , then p D, which is a contradiction to (n, D) = 1. | | The following statement is well known. Theorem 1.3. Let p be a prime > 2 with p ∤ DQ. Then: D (1) If = 1, then Up 1 0 mod p and Vp 1 2 mod p. Furthermore, p − ≡ − ≡ Q U p−1 0 mod p if and only if = 1. Moreover, V p−1 0 mod p if 2 ≡ p 2 ≡ Q and only if = 1. p − D (2) If = 1, then U 0 mod p and V 2Q mod p. Fur- p − p+1 ≡ p+1 ≡ Q thermore, U p+1 0 mod p if and only if = 1. Moreover, V p+1 0 2 ≡ p 2 ≡ Q mod p if and only if = 1. p − Notation 1.4. For an odd integer n > 1 with (n, D) = 1, we put ε(n) = D . n Definition 1.5.([2, sec.1]) An odd composite n is called a Lucas pseudo- prime with respect to (P,Q) if (n, D)=1 and Un ε(n) 0 mod n. − ≡ LUCASPSEUDOPRIMES 5 By Lemma 1.2, if n is a Lucas pseudoprime with respect to (P,Q), then n is prime to Q. Definition 1.6.([2, sec.3]) An odd composite n is called an Euler-Lucas Q pseudoprime with respect to (P,Q) if (n,DQ) = 1 and either = 1, n Q U n−ε(n) 0 mod n or = 1, V n−ε(n) 0 mod n. 2 ≡ n − 2 ≡ By the formula (4) in 1.1 and Lemma 1.2, we obtain the following asser- tion: Proposition 1.7. If n is an Euler-Lucas pseudoprime with respect to (P,Q), then n is a Lucas pseudoprime with respect to (P,Q). Lemma 1.8. Let p be a prime > 2 and p ∤ D. Let p ε(p) = 2sm with − (m, 2) = 1.Then U 0 mod p or V k 0 mod p for some k U 0 mod n αk βk mod n ak 1 mod n; k ≡ ⇔ ≡ ⇔ ≡ V 0 mod n αk βk mod n ak 1 mod n. k ≡ ⇔ ≡− ⇔ ≡− 2. Group schemes GD, U(D) and G(D) Throughout the sections hereafter, we fix a non-zero integer D. For an D odd integer n prime to D, we put ε(n)= . We adopt standard notations n in [13] concerning to affine group schemes. 6 NORIYUKI SUWA Definition 2.1. Let D Z, and put ∈ A = Z[t]/(t2 D), D − 1 G = G = Spec Z[X,Y, ]. D m,AD X2 DY 2 AYD/Z − The group law of GD is given by ∆ : (X,Y ) (X X + DY Y, X Y + Y X), 7→ ⊗ ⊗ ⊗ ⊗ ε : (X,Y ) (1, 0), 7→ X Y S : (X,Y ) , − . 7→ X2 DY 2 X2 DY 2 − − Furthermore, a homomorphism of affine group schemes 1 1 Nr : G = Spec Z[X,Y, ] G = Spec Z[T, ] D X2 DY 2 → m,Z T − is defined by 1 1 T X2 DY 2 : Z[T, ] Z[X,Y, ]. 7→ − T → X2 DY 2 − Put now U(D) = Ker[Nr : G G ]. More precisely, D → m,Z U(D) = Spec Z[X,Y ]/(X2 DY 2 1), − − and the group law of U(D) is given by ∆ : (X,Y ) (X X + DY Y, X Y + Y X), 7→ ⊗ ⊗ ⊗ ⊗ ε : (X,Y ) (1, 0), 7→ X Y S : (X,Y ) , − . 7→ X2 DY 2 X2 DY 2 − − 1 The group scheme U(D) is a torus over Z[ ]. 2D 1 In fact, put A = Z[√D, ]. Then an isomorphism of group schemes 2D over A 1 U(D) = Spec A[X,Y ]/(X2 DY 2 1) ∼ G = Spec A[U, ] A − − → m,A U is given by 1 U X + √DY : A[U, ] A[X,Y ]/(X2 DY 2 1). 7→ U → − − Furthermore the sequence of group schemes 0 U(D) G Nr G 0 −→ −→ D −→ m,Z −→ LUCASPSEUDOPRIMES 7 1 is exact over Z[ ]. 2D For the convenience, here is given a more concrete description of 2.1. 2.2. Let R be a commutative ring. Then we have 2 2 2 2 G (R) = (R[t]/(t D))× = (a, b) R ; a Db is invertible in R , D − { ∈ − } and the multiplication of GD(R) is given by (a, b)(a′, b′) = (aa′ + Dbb′, ab′ + a′b). The unit of GD(R) is given by (1, 0), and we have 1 a b (a, b)− = , . a2 Db2 −a2 Db2 − − Furthermore, for η = (a, b) G (R), we have ∈ D Nr(η)= a2 Db2, − and 2 2 2 U (R) = Ker[Nr : G (R) G (R)= R×]= (a, b) R ; a Db = 1 . D D → m { ∈ − } Notation 2.3. Let R be a commutative ring. We shall denote the unit (1, 0) U(D)(R) G (R) by I. For η = (a, b) G (R) and c R , we ∈ ⊂ D ∈ D ∈ × denote (ca, cb) GD(R) by cη. In particular, ( a, b) GD(R) is denoted by η. ∈ − − ∈ − Let ξ U(D)(R) and c R . Then cξ U(D)(R) if and only if c2 = 1. ∈ ∈ × ∈ 2.4. The correspondence c cI defines an embedding of multiplicative 27→ groups iR : R× (R[t]/(t D))×. The map iR is represented by the homomorphism of→ group schemes− 1 1 i : G = Spec Z[T, ] G = Spec Z[X,Y, ] m,Z T → D X2 DY 2 − defined by 1 1 (X,Y ) (T, 0) : Z[X,Y, ] Z[T, ]. 7→ X2 DY 2 → T − A homomorphism of group schemes 1 γ : G = Spec Z[X,Y, ] U(D) = Spec Z[X,Y ]/(X2 DY 2 1) D X2 DY 2 → − − − is defined by X2 + DY 2 2XY (X,Y ) , : 7→ X2 DY 2 X2 DY 2 − − 1 Z[X,Y ]/(X2 DY 2 1) Z[X,Y, ]. − − → X2 DY 2 − 8 NORIYUKI SUWA The sequence of group schemes i γ 0 G G U(D) 0 −→ m,Z −→ D −→ −→ 1 is exact over Z[ ]. Furthermore the composite of the embedding U(D) 2D → GD and γ : GD U(D) is the square map. → 1 An endomorphism σ of group schemes G = Spec Z[X,Y, ] is D X2 DY 2 defined by − 1 1 (X,Y ) (X, Y ) : Z[X,Y, ] Z[X,Y, ]. 7→ − X2 DY 2 → X2 DY 2 − − Let R be a ring and η = (a, b) GD(R). Then we have σ(η) = (a, b). By convention we denote σ(η) also∈ byη ¯. It is readily seen that ηη¯ = Nr(η−) I. Furthermore, we have 2 2 a + Db 2ab 1 2 1 γ(η)= , = Nr(η)− η = ηη¯− . a2 Db2 a2 Db2 − − Lemma 2.5. Let R be a commutative ring and η GD(R). Then: (1) γ(η)= I if and only if η¯ = η. ∈ (2) γ(η)= I if and only if η¯ = η. − − 1 Proof. The assertion is a direct consequence of the formula γ(η)= ηη¯− . Example 2.6. Let P , Q be integers = 0 with D = P 2 4Q = 0. Let 6 − 6 Un n 0, Vn n 0 denote the Lucas sequences associated to (P,Q). Then { } ≥ { } ≥ V U 1 k , k G (Z[ ]) 2 2 ∈ D 2Q since V 2 DU 2 = 4Qk. Moreover, we have k − k V U V U V U k , k l , l = k+l , k+l 2 2 2 2 2 2 1 in G (Z[ ]). In particular, D 2Q V U V U k P 1 k k , k = 1 , 1 = , . 2 2 2 2 2 2 Furthermore, we have V U V 2 + DU 2 2V U V U γ k , k = k k , k k = 2k , 2k , 2 2 4Qk 4Qk 2Qk 2Qk and therefore, V U V U k D + 2Q P k 2k , 2k = 2 , 2 = , 2Qk 2Qk 2Q 2Q 2Q 2Q LUCASPSEUDOPRIMES 9 1 in U(D)(Z[ ]). 2Q Lemma 2.7. Let P , Q and D be integers = 0, and let n be an odd integer > 1 with (n,Q) = 1 and P 2 4Q D mod6 n. Put − ≡ D + 2Q P ξ = , U(D)(Z/nZ). 2Q 2Q ∈ Then: (1) U 0 mod n if and only if ξk = I in U(D)(Z/nZ); k ≡ (2) V 0 mod n if and only if ξk = I in U(D)(Z/nZ). k ≡ − Proof. Put η = (P/2, 1/2) GD(Z/nZ). Then we have ξ = γ(η). Combin- ∈ k ing Lemma 2.5 and the equality η = (Vk/2,Uk/2) in GD(Z/nZ), we can obtain the result. Corollary 2.8. Under the notation above, we assume that (n,DQ) = 1. Then: n ε(n) (1) n is a Lucas pseudoprime with respect to (P,Q) if and only if ξ − = I in U(D)(Z/nZ); (2) n is an Euler-Lucas pseudoprime with respect to (P,Q) if and only if Q n−ε(n) Q n−ε(n) either = 1 and ξ 2 = I or = 1 and ξ 2 = I in n n − − U(D)(Z/nZ); (3) n is a strong Lucas pseudoprime with respect to (P,Q) if and only if either k ξm = I or ξ2 m = I for some k D (1) If = 1, then U(D)(F ) is a cyclic group of order p 1. p p − D (2) If = 1, then U(D)(F ) is a cyclic group of order p + 1. p − p D Proof. If = 1, then U(D) Z is isomorphic to the multiplicative p ⊗Z p group Gm,Zp . This implies the assertion. D On the other hand, if = 1, then we have p − U(D)(Zp)= α Zp[√D]× ; Nr (α) = 1 . { ∈ Qp(√D)/Qp } Let a pZ . Then we have ∈ p exp a√D 1+ pZ [√D] ∈ p and Nr (exp a√D) = 1. Qp(√D)/Qp Moreover the correspondence a exp a√D gives rise to an isomorphism of groups 7→ pZ ∼ Ker[U(D)(Z ) U(D)(F )]. p → p → p On the other hand, we have U(D)(Fp)= α Fp(√D)× ; Nr (α) = 1 . { ∈ Fp(√D)/Fp } Hence U(D)(Fp) is a cyclic group of order p + 1. Futhermore, since the quadratic extension Qp(√D)/Qp is unramified, the Teichm¨uller lifting gives a section of the reduction map U(D)(Z ) U(D)(F ). p → p Corollary 2.11.([1, Th.3.1]) Let p be a prime with (p, 2D) = 1 and α a positive integer. Then: D (1) If = 1, then U(D)(Z/pαZ) is a cyclic group of order (p 1)pα 1; p − − D (2) If = 1, then U(D)(Z/pαZ) is a cyclic group of order (p + 1)pα 1. p − − Notation 2.12.([1, Sec.2]) For a positive integer n with (n, 2D) = 1, we define 1 if n = 1 ϕD(n)= U(D)(Z/nZ) if n> 1. (| | LUCASPSEUDOPRIMES 11 If (n,m) = 1, then ϕD(nm)= ϕD(n)ϕD(m). Moreover, if p is a prime, then D (p 1)pα 1 if = 1 − − p ϕ (pα)= D D (p + 1)pα 1 if = 1. − p − Hence, if D is a square, ϕD(n) is nothing but the Euler function ϕ(n) for n with (n, 2D) = 1. Remark 2.13. (cf. [14]) Let D be an integer = 0. Then an affine group 6 scheme G(D) is defined by G = Spec Z[X,Y ]/(X2 DY 2 Y ) (D) − − with the group structure: ∆ : X X 1 + 1 X + 2DX Y + 2DY X, 7→ ⊗ ⊗ ⊗ ⊗ Y Y 1 + 1 Y + 2DY Y + 2X X, 7→ ⊗ ⊗ ⊗ ⊗ ε : X 0, Y 0, 7→ 7→ S : X X, Y Y. 7→ − 7→ The group scheme G(D) is smooth over Z. A surjective homomorphism of group schemes 1 β : G = Spec Z[X,Y, ] G = Spec Z[X,Y ]/(X2 DY 2 Y ) D X2 DY 2 → (D) − − − is defined by XY Y 2 X , Y : 7→ X2 DY 2 7→ X2 DY 2 − − 1 Z[X,Y ]/(X2 DY 2 Y ) Z[X,Y, ]. − − → X2 DY 2 − Moreover, 1 1 X T, Y 0 : Z[X,Y, ]/(XY,Y 2) Z[T, ] 7→ 7→ X2 DY 2 → T − gives rise to an isomorphism 1 G = Spec Z[T, ] m,Z T 1 ∼ Ker[β : G G ] = Spec Z[X,Y, ]/(XY,Y 2). → D → (D) X2 DY 2 − Furthermore, a homomorphism of affine group schemes α : G = Spec Z[X,Y ]/(X2 DY 2 Y ) (D) − − 12 NORIYUKI SUWA U(D) = Spec Z[X,Y ]/(X2 DY 2 1) → − − is defined by X 2DY + 1, Y 2X : 7→ 7→ Z[X,Y ]/(X2 DY 2 1) Z[X,Y ]/(X2 DY 2 Y ). − − → − − 1 Hence α : G U(D) is an isomorphism over Z[ ]. (D) → 2D The homomorphism γ : GD U(D) is nothing but the composite of β : G G and α : G U→(D). D → (D) (D) → It would be suitable to define the function ϕD(n) by 1 if n = 1 ϕD(n)= G (Z/nZ) if n> 1. (| (D) | since the reduction map G (Z ) G (Z/pαZ) is surjective for any (D) p → (D) prime p and any α 1. This definition respects Arnault [1] since the ≥ homomorphism α : G(D) U(D) is isomorphic over Z/nZ with (n, 2D) = 1. → α α Moreover, if p is a prime divisor of 2D, then we have ϕD(p )= p . It is also verified that: α α 1 (1) If 2 ∤ D, then G(D)(Z/2 Z) is isomorphic to Z/2Z Z/2 − Z; α × α 1 (2) If D 3 mod 9, then G(D)(Z/3 Z) is isomorphic to Z/3Z Z/3 − Z; ≡− α × α (3) If p is a prime divisor of D, then G(D)(Z/p Z) is isomorphic to Z/p Z except the case (2). 3. Statement of the theorem Notation 3.1. Let D be an integer = 0 and p a prime > 2 with (p, D) = 1. 6 ξ For ξ U(D)(F ), we define a symbol by ∈ p p h p−iε(p) ξ 1 if ξ 2 = I, = p−ε(p) p ( 1 if ξ 2 = I. h i − − We have gotten an exact sequence square [ p ] 1 I U(D)(F ) U(D)(F ) 1 1. −→ {± } −→ p −→ p −→ {± } −→ since U(D)(Fp) is a cyclic group of order p ε(p). Moreover, let n be an odd integer > 1 with (n, D) = 1. For ξ U(−D)(Z/nZ), we define a symbol ξ ∈ by n h i ξ ξ ordpn = . n p p n h i Y| h i LUCASPSEUDOPRIMES 13 Example 3.2. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Let P, Q Z. Assume6 that (n,Q) = 1 and P 2 4Q D mod n. Put ∈ − ≡ D + 2Q P ξ = , . 2Q 2Q As is noticed in Example 2.6, we have V U ξk = 2k , 2k . 2Qk 2Qk Hence Theorem 1.3 implies that ξ Q = p p h i for each prime divisor p of n, and therefore, ξ Q = . n n h i We shall introduce notations after Monier [9]. Definition 3.3. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Let n ε(n) = 2sm with (m,6 2) = 1. We put: − n ε(n) B˜ (n, D)= ξ U(D)(Z/nZ) ; ξ − = I , ℓpsp { ∈ } ξ n−ε(n) = 1 and ξ 2 = I, B˜ (n, D)= ξ U(D)(Z/nZ) ; n , eℓpsp ∈ h iξ n−ε(n) or = 1 and ξ 2 = I n − − h i k B˜ (n, D)= ξ U(D)(Z/nZ) ; ξm = I, or ξ2 m = I for some k We denote often B˜ℓpsp(n, D), B˜eℓpsp(n, D), B˜sℓpsp(n, D) by B˜ℓpsp, B˜eℓpsp, B˜sℓpsp, respectively, when (n, D) is fixed. Now we can state the main theorem. Theorem 3.4. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Let r denote the number of6 distinct prime divisors of n, and put s n ε(n) = 2 m with (m, 2) = 1 and ν = min ord2(p ε(p)). Then: − p n − | (1) B˜ B˜ B˜ . ℓpsp ⊃ eℓpsp ⊃ sℓpsp (2) B˜ℓpsp is a subgroup of U(D)(Z/nZ) and B˜ = (n ε(n),p ε(p)); | ℓpsp| − − p n Y| 14 NORIYUKI SUWA (3) B˜eℓpsp is a subgroup of U(D)(Z/nZ) and n ε(n) 2 − ,p ε(p) if s = ν 2 − p n Y| if s>ν and n ε(n) − ,p ε(p) ordpn 0 mod2 for any ˜ 2 − ≡ Beℓpsp(n, D) = p n p n with ord (p ε(p)) 1 2 2 s˜ : U(D)Z[ 1 ] = Spec Z[ ][X,Y ]/(X DY 1) 2D 2D − − 1 1 ∼ Gm,Z[ 1 ] = Spec Z[ ][U, ] −→ 2D 2D U is defined by 1 1 1 U X + dY : Z[ ][U, ] Z[ ][X,Y ]/(X2 DY 2 1). 7→ 2D U → 2D − − Then for an odd integer n with (n, D) = 1, an isomorphism of groupss ˜ : U(D)(Z/nZ) ∼ (Z/nZ)× is given by ξ = (a, b) a+db. Moreovers ˜ induces isomorphisms→ 7→ B˜ (n, D) ∼ B (n), ℓpsp −→ psp B˜ (n, D) ∼ B (n) eℓpsp −→ epsp and a bijection B˜ (n, D) ∼ B (n). sℓpsp → spsp LUCASPSEUDOPRIMES 15 Here n 1 B (n)= a (Z/nZ)× ; a − = 1 , psp { ∈ } n−1 a B (n)= a (Z/nZ)× ; a 2 = , epsp { ∈ n } m 2km B (n)= a (Z/nZ)× ; a = 1, or a = 1 for some k Lemma 4.2. Let n be an odd integer > 1 and k an integer 1. Put ν = ≥ 2k min ord2(p ε(p)). Then there exists ξ U(D)(Z/nZ) such that ξ = I p n − ∈ − if| and only if k ν 1. ≤ − e1 e2 er Proof. Put n = p1 p2 pr , where p1,p2,...,pr are distinct primes. Let k k · · · 2 2 ei ξ U(D)(Z/nZ). Then we have ξ = I if and only if ξ I mod pi k ∈ ei− 2 ≡− for each i. There exists ξi U(D)(Z/pi Z) such that ξi = I if and only if k∈+1 ei − pi ε(pi) is divisible by 2 , since U(D)(Z/pi Z) is a cyclic group of order − e ϕ (p i ) = (p ε(p ))pei 1 and 2 is prime to p . Hence we obtain the result. D i i − i − i Lemma 4.3. Let n be an odd integer > 1. Put s = ord (n ε(n)) and 2 − ν = min ord2(p ε(p)). Then we have s ν.Furthermore, p n − ≥ | s = ν ord n 1 mod2, ⇔ p ≡ p n X| ord2(p ε(p))=ν − s>ν ord n 0 mod2. ⇔ p ≡ p n X| ord2(p ε(p))=ν − Proof. Let p be a prime divisor of n. Noting that ord (p ε(p)) = ν p ε(p) + 2ν mod 2ν+1, 2 − ⇔ ≡ ord (p ε(p)) >ν p ε(p) mod2ν+1, 2 − ⇔ ≡ we obtain n ε(p)ordpn + ε(p)ord n 2ν ≡ p p n p n Y| X| ord2(p ε(p))=ν − ε(n)+ ord n 2ν mod 2ν+1. ≡ p p n X| ord2(p ε(p))=ν − Corollary 4.4. Let n be an odd integer > 1. Put s = ord (n ε(n)) and 2 − ν = min ord2(p ε(p)). Then: p n − | (1) For any prime divisor p of n we have s ord2(p ε(p)) if and only if s = ν; ≤ − (2) There exists a prime divisor p of n such that s > ord2(p ε(p)) if and only if s>ν. − LUCASPSEUDOPRIMES 17 Corollary 4.5. Let D be an integer = 0 and n an odd integer > 1 with 6 (n, D) = 1. Put s = ord2(n ε(n)) and ν = min ord2(p ε(p)). Let − p n − | ξ U(D)(Z/nZ). Then: ∈ ν−1 ξ (1) If ξ2 = I, then = 1; n ν−1 h i ξ (2) If ξ2 = I and s>ν, then = 1; − n ν−1 h ξ i (3) If ξ2 = I and s = ν, then = 1. − n − ν−1 Proof. First assume that ξ2 = Ih. Theni for each prime divisor p of n we have − − p ε(p) 2ν−1 p ε(p) ξ 2 = (ξ ) 2ν I mod p, ≡ ξ which implies that = 1. Hence we obtain the first assertion. p h i ν−1 Now we assume that ξ2 = I. Then we have − p−ε(p) I mod p if ord2(p ε(p)) >ν ξ 2 − ≡ I mod p if ord (p ε(p)) = ν, (− 2 − and therefore, ξ 1 if ord (p ε(p)) >ν = 2 − p ( 1 if ord2(p ε(p)) = ν. h i − − Hence we obtain ξ = ( 1)e, n − where h i e = ordpn. p n X| ord2(p ε(p))=ν − Therefore the last two assertions follow from Lemma 4.3. 4.6. Proof of Theorem 3.4. n−ε(n) (1) Assume first that ξ B˜ . Then by definition we have ξ 2 = I, ∈ eℓpsp ± which implies that ξn ε(n) = I, that is, ξ B˜ . − ∈ ℓpsp Assume now that ξ B˜sℓpsp. Then by definition and by Lemma 4.2, we k ∈ have ξm = I or ξ2 m = I for some k ν 1. − ≤ − Case (a): s = ν. Then we have − n ε(n) 2ν−1m ξ 2 = ξ = I. ± 18 NORIYUKI SUWA Furthermore, by Corollary 4.5 we have ν−1 ξ ξm 1 if ξ2 m = I = = 2ν−1m n n ( 1 if ξ = I, h i h i − − since m is odd. This implies that ξ B˜ . ∈ eℓpsp Case (b): s>ν. We have − n ε(n) 2s−1m 2ν−1m 2s−ν ξ 2 = ξ = (ξ ) = I. ξ On the other hand, by Corollary 4.5, we have = 1, which implies that n ξ B˜ . h i ∈ eℓpsp (2) Applying Lemma 4.1 to d = n ε(n), we obtain the assertion. − n−ε(n) (3) Put C = ξ U(D)(Z/nZ) ; ξ 2 = I . Then C is a subgroup of U(D)(Z/nZ).{ Furthermore,∈ by Lemma 4.1 we} have n ε(n) C = − ,p ε(p) . | | 2 − p n Y| ν−1 Case (a): s = ν. If ξ C, then ξ2 m = I. Hence by Corollary 4.5, we ξ ξm ∈ obtain = = 1, which implies that ξ B˜ n n ∈ eℓpsp On theh otheri h hand,i by Lemma 4.2 there exists ξ U(D)(Z/nZ) such that ∈ ν−1 ξ ξ2 = I. Then it follows from Corollary 4.5 that = 1 and that − n − ξ B˜ (n, D). h i ∈ eℓpsp ξ It follows that the homomorphism B˜ 1 defined by ξ is eℓpsp → {± } 7→ n surjective and that Ker[B˜ 1 ]= C, and therefore C is a subgrouph i eℓpsp → {± } of B˜eℓpsp of index 2. Case (b): s>ν. By Lemma 4.2, there does not exist ξ U(D)(Z/nZ) such n−ε(n) − ∈ 2s 1m that ξ 2 = ξ = I, which implies that B˜ C. − eℓpsp ⊂ Now let ξ C. Let p be a prime divisor of n. If ord2(p ε(p)) s, then ξ ∈ − ≥ = 1.Indeed, we have p h i − − − p ε(p) p ε(p) m 2s−1m p ε(p) ξ 2 (ξ 2 ) = (ξ ) 2s I mod p ≡ ≡ n−ε(n) since ξ 2 I mod p and (m, 2) = 1. ≡ Hence, if ordpn 0 mod 2 for any prime divisor p of n with ord2(p ≡ ξ − ε(p)) On the other hand, assume that there exists a prime divisor p of n such e1 e2 er that ord2(p ε(p)) k ξ U(D)(Z/nZ) such that ξ2 = I if and only if k + 1 ν. Hence, if ∈ − ≤ k ν, then B = ø. Hence we obtain a partition of B˜ ≥ k sℓpsp B˜sℓpsp = C0 B0 B1 Bν 1, ∪ ∪ ∪···∪ − and therefore r r(ν 1) B˜ =(1+1+2 + + 2 − ) (m,p ε(p)) | sℓpsp| · · · − p n Y| 2rν 1 = 1+ − (m,p ε(p)). 2r 1 − p n − Y| Remark 4.7. Under the notations of Theorem 3.4, let C˜ denote the sub- group of U(D)(Z/nZ) generated by B˜sℓpsp. As is verified in 4.6, we have B˜sℓpsp = C0 B0 B1 Bν 1 and Cν 1 C0 B0 B1 Bν 2. Hence we obtain∪ ∪ ∪···∪ − − ⊃ ∪ ∪ ∪···∪ − C˜ = Cν 1 Bν 1, − ∪ − and therefore r(ν 1)+1 C˜ = 2 Cν 1 = 2 − (m,p ε(p)). | | | − | − p n Y| In particular, B˜sℓpsp is a subgroup of U(D)(Z/nZ) if and only if Cν 1 = − C0 B0 B1 Bν 2. This is the case only when r =1 or ν = 1. ∪ ∪ ∪···∪ − e1 e2 er Remark 4.8. Let n = p1 p2 pr be an odd number, where p1,p2,...,pr · · · s are distinct primes. Put n ε(n) = 2 m with (m, 2) = 1 and pi ε(pi) = si − − 2 mi with (mi, 2) = 1 for each i, and put ν = min si. Then we have 1 i r ≤ ≤ r ord2 ϕD(n)= si Xi=1 and, by Theorem 3.4, r ord B˜ = min(s,s ), 2| ℓpsp| i Xi=1 1+ r(s 1) if s = ν − r if s>ν and ei 0 mod2 ˜ min(s 1,si) ≡ ord2 Beℓpsp = − for any i with s ord C˜ =1+ r(ν 1). 2| | − r Furthermore, B˜ , B˜ and C˜ are equal to (m,m ) up to powers | ℓpsp| | eℓpsp| | | i i=1 of 2. Y 5. Some consequences of the theorem Proposition 5.1. Let D be an integer = 0 and n an odd integer > 1 with 6 (n, D) = 1. Then B˜ℓpsp(n, D) = B˜eℓpsp(n, D) if and only if either n is a prime power, or n is a square and ord2(p ε(p)) < ord2(n ε(n)) for each prime divisor p of n. − − Proof. By Remark 4.8, B˜ = B˜ up to powers of 2. It follows that | ℓpsp| | eℓpsp| B˜ℓpsp = B˜eℓpsp if and only if ord2 B˜ℓpsp = ord2 B˜eℓpsp . Put now n = pe1 pe2 per , where| p| ,p ,...,p| are| distinct primes, and 1 2 · · · r 1 2 r n ε(n) = 2sm with (m, 2) = 1. Put p ε(p ) = 2si m with (m , 2) =1 for − i − i i i each i, and ν = min si. 1 i r ≤ ≤ Case (a): s = ν. By Remark 4.8 we have r ord B˜ = min(s,s )= rs, 2| ℓpsp| i i=1 r X ord B˜ =1+ min(s 1,s )=1+ r(s 1). 2| eℓpsp| − i − Xi=1 Hence ord B˜ = ord B˜ if and only if r = 1. 2| ℓpsp| 2| eℓpsp| Case (b): s>ν and ei 0 mod 2 for any i with si Corollary 5.2. Let D be an integer = 0 and n an odd composite with 6 (n, D) = 1. Then B˜ ϕ (n)/2. | eℓpsp|≤ D Proof. It is sufficient to prove that B˜eℓpsp = U(D)(Z/nZ) since B˜eℓpsp is a subgroup of U(D)(Z/nZ). If n is not a Carmichael-Lucas6 number to base D, then B˜ B˜ < ϕ (n). On the other hand, if n is a Carmichael- | eℓpsp| ≤ | ℓpsp| D Lucas number to base D, then B˜eℓpsp < B˜ℓpsp since n is neither a prime power nor a square. | | | | Lemma 5.3. Let D be an integer = 0 and n an odd integer > 1 with 6 (n, D) = 1. Put s = ord2(n ε(n)) and ν = min ord2(p ε(p)). Let C˜ − p n − | denote the subgroup of U(D)(Z/nZ) generated by B˜sℓpsp. Then C˜ = B˜eℓpsp if and only if either s = ν, or n is a prime power, or n = pαqβ, where p, q are distinct primes with ord2(p ε(p)) = ord2(q ε(q)) and α β 1 mod 2. − − ≡ ≡ Proof. By Remark 4.8, C˜ = B˜ up to powers of 2. It follows that | | | eℓpsp| B˜eℓpsp = C˜ if and only if ord2 B˜eℓpsp = ord2 C˜ . Put now n = pe1 pe2 per ,| where| p ,p ,...,p| | are distinct primes, and 1 2 · · · r 1 2 r n ε(n) = 2sm. Put p ε(p ) = 2si m with (m , 2) = 1 for each i. − i − i i i Case (a): s = ν. By Remark 4.8 we have ord B˜ =1+ r(ν 1) = ord C˜ =1+ r(ν 1). 2| eℓpsp| − 2| | − Case (b): s>ν and ei 0 mod 2 for any i with si Proposition 5.4. Let D be an integer = 0 and n an odd integer > 1 with 6 (n, D) = 1. Then B˜ = B˜ if and only if either n ε(n) 2 mod4, eℓpsp sℓpsp − ≡ or n is a prime power, or n = pαqβ, where p,q are distinct primes with p ε(p) q ε(q) 2 mod4 and α β 1 mod2. − ≡ − ≡ ≡ ≡ Proof. Let C˜ be a subgroup of U(D)(Z/nZ) generated by B˜sℓpsp. Then B˜eℓpsp = B˜sℓpsp if and only if B˜eℓpsp = C˜ and C˜ = B˜sℓpsp. Therefore, combining Lemma 5.3 and Remark 4.7, we obtain the result. Proposition 5.5. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Then B˜ = ϕ (n)/2 if and6 only if n is a Carmichael-Lucas | eℓpsp| D number to base D and ord2(p ε(p)) < ord2(n ε(n)) for each prime divisor p of n. − − Proof. Assume that B˜eℓpsp = ϕD(n)/2. Then it follows from Theorem 3.4 that n is square-free.| Put |n = p p p , where p ,p ,...,p are distinct 1 2 · · · r 1 2 r primes, and n ε(n) = 2sm with (m, 2) = 1. Put p ε(p ) = 2si m with − i − i i (mi, 2) = 1 for each i, and ν = min si. Then again by Theorem 3.4 we have 1 i r r 2 and m m for each i. ≤ ≤ ≥ i| Case (a): s = ν. By Remark 4.8, we have r ord B˜ =1+ r(s 1), ord ϕ (n)= s , 2| eℓpsp| − 2 D i Xi=1 which implies that r 1+ r(s 1) = 1+ s , − − i Xi=1 and therefore r (s s + 1) = 2. i − Xi=1 Since si s + 1 1 for each i, we have r = 2 and s1 = s2 = s. It follows from Lemma− 4.3≥ that s>ν, which contradicts to s = ν. Case (b): s>ν. By Remark 4.8 we have r ord B˜ = 1+ min(s 1,s ) 2| eℓpsp| − − i Xi=1 since n is square-free. This implies that r r 1+ min(s 1,s )= 1+ s , − − i − i Xi=1 Xi=1 24 NORIYUKI SUWA and therefore r r min(s 1,s )= s . − i i Xi=1 Xi=1 It follows that si s 1 for each i. Hence it is concluded by Corollary 3.9 that n is a Carmichael-Lucas≤ − number to base D, since m m for each i. i| Conversely, let n = p1p2 pr be a Carmichael-Lucas number to base D, where p ,p ,...,p are distinct· · · primes. Assume that we have ord (p 1 2 r 2 i − ε(pi)) < ord2(n ε(n)) for each i. Then by Corollary 3.9 we obtain (pi n ε(n) − − ε(p )) − for each i. It follows from Theorem 3.4 that i | 2 r r 1 n ε(n) 1 ϕ (n) B˜ = − ,p ε(p )) = (p ε(p )) = D | eℓpsp| 2 2 i − i 2 i − i 2 i=1 i=1 Y Y since s>ν = min si and n is square-free. 1 i r ≤ ≤ Example 5.6. When D = 5, the first three examples with B˜ = | eℓpsp| ϕD(n)/2 are given by 323 = 17 19, 6721 = 11 13 47 and 11663 = 107 109. × × × × Corollary 5.7. Let D be a non-square and n an odd composite with (n, D)= 1. Then B˜sℓpsp ϕD(n)/2. In particular, B˜sℓpsp = ϕD(n)/2 if and only if n = pq is| a Carmichael-Lucas|≤ number to base| D with| q = p + 2, ε(p)= 1, ε(q) = 1, p 1 mod4. − ≡ Proof. Combining Propositions 5.5 and 5.4, we obtain the result. Example 5.8. When D = 5, the first three examples with B˜ = | sℓpsp| ϕD(n)/2 are given by 323 = 17 19, 19043 = 137 139 and 39203 = 197 199. × × × Remark 5.9. Let D be a square and n an odd composite with (n, D) = 1 and n = 9. Then, as is well known, B˜ ϕ(n)/4 ([9], [10]). In 6 | sℓpsp| ≤ particular, B˜ = ϕ(n)/4 if and only if either n = pq, where p, q are | sℓpsp| primes with p 3 mod4 and q = 2p 1, or n = p1p2p3 is a Carmaichael number, where≡p ,p ,p are different primes− with p p p 3 mod 4. 1 2 3 1 ≡ 2 ≡ 3 ≡ Remark 5.10. Let D be an integer = 0, n an odd integer > 1 with s 6 (n, D)=1. Put n ε(n) = 2 m with (m, 2) = 1 and ν = min ord2(p ε(p)). − p n − Then: | (1) B˜eℓpsp = I if and only if either n is a power of 3 and D 1 mod 3, or n ε(n) {±2} mod4and(m,p ε(p)) = 1 for each prime divisor≡ p of n, − ≡ − LUCASPSEUDOPRIMES 25 or n = pαqβ, where p, q are distinct primes with p ε(p) q ε(q) 2 mod 4, (m,p ε(p)) = (m,q ε(q)) = 1 and α β −1 mod≡ 2.− ≡ − − ≡ ≡ (2) B˜sℓpsp = I if and only if ν = 1 and (m,p ε(p)) = 1 for each prime divisor p of n{±. } − Hereafter we mention several remarks with relation to preceding works [1], [2] and [15]. Definition 5.11. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. We put: 6 Bℓpsp(n, D)= there exist Q Z/nZ such that P 2 4Q = D and P Z/nZ ; ∈ − , ∈ n is a Lucas pseudoprime with respect to (P,Q) Beℓpsp(n, D)= there exists Q (Z/nZ) such that P 2 4Q = D and P Z/nZ ; ∈ × − , ∈ n is an Euler-Lucas pseudoprime with respect to (P,Q) Bsℓpsp(n, D)= there exists Q Z/nZ such that P 2 4Q = D and P Z/nZ ; ∈ − . ∈ n is a strong Lucas pseudoprime with respect to (P,Q) Notation 5.12. Let D be an integer = 0. Put 6 1 V = Spec Z[T, ]. D T 2 D − 1 Then VD is an open subscheme of the affine line AZ over Z. A morphism of affine schemes 1 π : V = Spec Z[T, ] U(D) = Spec Z[X,Y ]/(X2 DY 2 1) D T 2 D → − − − is defined by T 2 + D 2T 1 X , Y : Z[X,Y ]/(X2 DY 2 1) Z[T, ]. 7→ T 2 D 7→ T 2 D − − → T 2 D − − − Moreover, put 1 V˜ = Z[X,Y, ]/(X2 DY 2 1). D X 1 − − − V˜D is an open subscheme of U(D). Moreover, π induces an isomorphism ˜ πZ[ 1 ] : VD,Z[ 1 ] ∼ VD,Z[ 1 ] 2D 2D → 2D 26 NORIYUKI SUWA 1 over Z[ ]. In fact, the correspondence 2D DY 1 1 T : Z[T, ] Z[X,Y, ]/(X2 DY 2 1) 7→ X 1 T 2 D → X 1 − − − − − defines the inverse of π 1 . Z[ 2D ] Example 5.13. Let D be an integer = 0, n an odd integer > 1 with 6 2 (n, D) = 1 and P VD(Z/nZ). Put Q = (P D)/4 Z/nZ. Then Q is invertible in Z/nZ,∈ and − ∈ P 2 + D 2P D + 2Q P π(P )= , = , . P 2 D P 2 D 2Q 2Q − − Lemma 5.14. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Then the morphism π : V 6 U(D) induces bijections D → B (n, D) ∼ V˜ (Z/nZ) B˜ (n, D), ℓpsp → D ∩ ℓpsp B (n, D) ∼ V˜ (Z/nZ) B˜ (n, D), eℓpsp → D ∩ eℓpsp B (n, D) ∼ V˜ (Z/nZ) B˜ (n, D). sℓpsp → D ∩ sℓpsp Proof. The assertion follows from Corollary 2.8 and Example 5.13. Lemma 5.15. Let D be an integer = 0, n an odd integer > 1 with (n, D) = 1 6 and ξ U(D)(Z/nZ). Then ξ VD ξ I mod p for each prime divisor p of n.∈ ∈ ⇔ 6≡ Proof. Let ξ U(D)(Z/nZ). Then, by definition, ξ V˜D(Z/nZ) if and only if a 1 is invertible∈ in Z/nZ, which means that a 1∈ mod p for each prime divisor− p of n. On the other hand, a 1 mod p if6≡ and only if ξ I mod p, since a2 Db2 1 mod p. ≡ ≡ − ≡ Corollary 5.16. Let D be an integer = 0 and n an odd integer > 1 with 6 (n, D) = 1. Then the group U(D)(Z/nZ) is genarated by V˜D(Z/nZ). Proof. First let p be a prime and θ a generator of U(D)(Z/pαZ). Observe that, except in the case of p = 3 and ε(p) = 1, we have ξ I mod p. 6≡ ± α On the other hand, if p = 3 and ε(p) = 1, then we have V˜D(Z/p Z) = α α ξ U(D)(Z/p Z) ; ξ I mod p , and therefore, V˜D(Z/p Z) = {U(D∈)(Z/pαZ) /2. ≡ − } | | | Put now n =| pe1 pe2 per , where p ,p ,...,p are distinct primes. Then 1 2 · · · r 1 2 r the correspondence ξ (ξ mod pe1 ,ξ mod pe2 ,...,ξ mod per ) gives rise to an isomorphism 7→ e e U(D)(Z/nZ) ∼ U(D)(Z/p 1 Z) U(D)(Z/p 2 Z) U(D)(Z/per Z). −→ 1 × 2 ×···× r LUCASPSEUDOPRIMES 27 Let H denote the subgroup of U(D)(Z/nZ) genarated by V˜D(Z/nZ). Take ei a generator θi of U(D)(Z/pi Z). Except in the case of pi = 3 and ε(pi) = 1, we have ( I,..., I, θ , I,..., I) V˜ (Z/nZ), − − − i − − ∈ D which implies (I,...,I,θ ,I,...,I)= ( I,..., I, θ , I,..., I) H. i − − − − i − − ∈ Hence we obtain U(D)(Z/nZ) : H < 2, and therefore U(D)(Z/nZ)= H. | | Corollary 5.17. Let D be an integer = 0 and n an odd integer > 1 with (n, D) = 1. Then: 6 (1) B˜ℓpsp(n, D)= U(D)(Z/nZ) if and only if Bℓpsp(n, D)= VD(Z/nZ). (2)([15, Th.7]) B (n, D) = V (Z/nZ). eℓpsp 6 D Remark 5.18. Williams [15] defines an odd composite n to be a Carmichael- Lucas number to base D if n is prime to D and Bℓpsp(n, D) = VD(Z/nZ). The first assertion of Corollary 5.17 assures that Definition 3.8 is equivalent to Williams’. Remark 5.19. Lemma 5.14 allows us to deduce formulas for B , B | ℓpsp| | eℓpsp| and B from those for B˜ , B˜ and B˜ , respectively. This | sℓpsp| | ℓpsp| | eℓpsp| | sℓpsp| is done indeed by Baillie and Wagstaff [2, Th.2] for Bℓpsp, and by Arnault [1, Th.1.5] for Bsℓpsp. 6. Frobenius pseudoprime Grantham ([4], [5]) defines the notion of Frobenius pseudoprimes and strong Frobenius pseudoprimes with respect to a polynomial f(t) Z[t]. We adopt here the definition given in [3] for a quadratic polynomial.∈ Definition 6.1.[3, Def.3.6.5] Let P , Q be integers = 0, and put D = P 2 4Q. Assume that D is not a square. An odd composite6 n is called a Frobenius− pseudoprime with respect to (P,Q) if n is prime to QD and t if ε(n) = 1 tn = P t if ε(n)= 1 ( − − in the residue ring Z[t]/(n,t2 Pt + Q). − P + √D Remark 6.2. The correspondence t gives rise to an isomor- 7→ 2 phism of rings 1 1 Z[ ][t]/(t2 Pt + Q) ∼ Z[ ][√D]. 2 − −→ 2 28 NORIYUKI SUWA P √D Under this identification, we have P t = − . This implies the − 2 following assertion. Let n be an odd composite with (n,DQ) = 1, and put η = (P/2, 1/2) ∈ GD(Z/nZ). Then n is a Frobenius pseudoprime with respect to (P,Q) if and only if η if ε(n) = 1 ηn = η¯ if ε(n)= 1. ( − Proposition 6.3. Let P , Q be integers = 0 with D = P 2 4Q not a square, and let n be an odd composite with 6 (n,DQ) = 1. Assume− that n is a Frobenius pseudoprime with respect to (P,Q). Then: (1)([5, Th.4.9]) n is a Lucas pseudoprime with respect to (P,Q). (2) n is a pseudoprime to base Q. Proof. Put P 1 D + 2Q P η = , and ξ = γ(η)= , . 2 2 2Q 2Q n (1) In the case of ε(n) = 1, we have η = η in G(D)(Z/nZ), and therefore n n 1 ξ = ξ in U(D)(Z/nZ), which implies ξ − = I. On the other hand, in the case of ε(n) = 1, we have ηn =η ¯ in n 1 − 1 G(D)(Z/nZ). Hence we obtain ξ = ξ− in U(D)(Z/nZ) since γ(¯η) = ξ− . This implies ξn+1 = I. (2) In both the cases we obtain Qn = Q in Z/nZ from the fact Nr(η) = Nr(¯η)= Q. Remark 6.4. The second assertion of Proposition 6.3 is a special case of Grantham [5, Th. 4.3]) . Remark 6.5. Let Un n 0, Vn n 0 denote the Lucas sequences associated to (P,Q). Then we{ have} ≥ { } ≥ V U P 1 k k , k = , 2 2 2 2 1 in G (Z[ ]), as is remarked in Example 2.6. This implies the following (D) 2Q assertion. Let n be an odd composite with (n,DQ) = 1. Then n is a Frobenius pseudoprime with respect to (P,Q) if and only if 2 if ε(n) = 1 Un ε(n) 0 mod n and Vn ε(n) − ≡ − ≡ 2Q if ε(n)= 1 − LUCASPSEUDOPRIMES 29 ([3, Th.3.6.6]). Remark 6.6. Let P , Q be integers = 0 with D = P 2 4Q not a square, and let n be an odd composite with6 (n,DQ) = 1. Put−η = (P/2, 1/2) ∈ G (Z/nZ). Then γ(η) B˜ (n, D) and Q = Nr(η) B (n) if and only D ∈ ℓpsp ∈ psp if ηn ε(n) = aI, where a Z/nZ with − ∈ 1 if ε(n) = 1 a2 = Q2 if ε(n)= 1. ( − n ε(n) n ε(n) In fact, if γ(η) B˜ℓpsp(n, D), then γ(η − )= I, and therefore, η − ∈ 2 n ε(n) n ε(n) = aI for some a (Z/nZ)×. It follows that a = Nr(η − ) = Q − = n 1 1 ε(n) ∈ 2 1 ε(n) Q − Q − . Hence, if Q Bpsp(n), then we obtain a = Q − . ∈ n ε(n) Conversely, assume that there exists a Z/nZ such that η − = aI and 2 1 ε(n) n ε(n)∈ n ε(n) n ε(n) a = Q − . Then we obtain γ(η) − = I and Q − = Nr(η) − = 2 n 1 a , and therefore Q − = 1. Proposition 6.7. Let P , Q be integers = 0 with D = P 2 4Q not a square, and let n be an odd composite with6 (n,DQ) = 1. Assume− that n is a Frobenius pseudoprime with respect to (P,Q). Then n is an Euler pseudoprime to base Q if and only if n is an Euler-Lucas pseudoprime with respect to (P,Q). 1 Proof. Put η = (P/2, 1/2) GD(Z/nZ) and ξ = γ(η) = ηη¯− . Then we have ∈ η if ε(n) = 1 ηn = η¯ if ε(n)= 1 ( − since n is a Frobenius pseudoprime with respect to (P,Q). Assume that n n−1 Q is an Euler pseudoprime to base Q, that is to say, Q 2 mod n. ≡ n − n 1 Q n 1 Then, by Remark 2.9, we have ξ 2 = η . If ε(n) = 1, then we n − obtain n−ε(n) Q ξ 2 = I. n If ε(n)= 1, then we obtain − − n ε(n) Q n 1 Q n 1 1 Q n 1 Q ξ 2 = η − ξ = η − (ηη¯− )= η η¯− = I. n n n n Therefore n is an Euler-Lucas pseudoprime with respect to (P,Q ). Conversely, assume that n is an Euler-Lucas pseudoprime with respect to (P,Q). Then we have n−ε(n) Q ξ 2 = I. n 30 NORIYUKI SUWA If ε(n)= 1, we have − − n 1 1 n+1 Q Q n 1 ξ 2 (ηη¯− )= ξ 2 = I = η η¯− . n n − n 1 Q n 1 Hence in both the cases, we obtain ξ 2 = η . It follows from n − n−1 Q Remark 2.9 that Q 2 mod n. ≡ n Remark 6.8. The if-part of Proposition 6.7 is proved by Baillie and Wagstaff [2, Th.5]. Corollary 6.9. Let P be an integers = 0, Q = 1 with D = P 2 4Q not a square, and let n be an odd composite6 with (n,± D) = 1. Then n−is a Frobenius pseudoprime with respect to (P,Q) if and only if n is an Euler- Lucas pseudoprime with respect to (P,Q). Proof. Assume that n is a Frobenius pseudoprime with respect to (P,Q). It follows from Proposition 6.7 that n is an Euler-Lucas pseudoprime with respect to (P,Q) since n is an Euler pseudoprime to base Q = 1. Conversely, assume that n is an Euler-Lucas pseudoprime with± respect to 1 (P,Q). Put η = (P/2, 1/2) GD(Z/nZ) and ξ = ηη¯− . As is remarked at the end of 2.4, we have ∈ I if Q = 1 ηη¯ = Nr(η)I = I if Q = 1, (− − and therefore ξ if Q = 1 η2 = ξ if Q = 1. (− − n−ε n n ε(n) ( ) In the case of Q = 1, we have η − = ξ 2 = I by Corollary 2.8, which implies η if ε(n) = 1 ηn = ηε(n) = η¯ if ε(n)= 1. ( − n−ε n n ε(n) ( ) In the case of Q = 1 and n 1 mod 4, we have η − = ( ξ) 2 = n−ε(n) − ≡ − ( 1) 2 I by Corollary 2.8, which implies − − n n ε(n) ε(n) η if ε(n) = 1 η = ( 1) 2 η = − ( η¯) =η ¯ if ε(n)= 1. (− − − LUCASPSEUDOPRIMES 31 n−ε n n ε(n) ( ) In the case of Q = 1 and n 3 mod 4, we have η − = ( ξ) 2 = n−ε(n) − ≡ − ( 1) 2 ( I) by Corollary 2.8, which implies − − − n n ε(n) +1 ε(n) η if ε(n) = 1 η = ( 1) 2 η = − ( η¯) =η ¯ if ε(n)= 1. (− − − ackowledgement This article was finished during the author’s stay at Torre Archimede of Universit`adegli studi di Padova. He would like to express his hearty thanks to its hospitality. In particular he is very grateful to Marco Garuti and Alessandro Languasco for their helpful advices and notices, often accompa- nied with caff`enormale. Finally he highly appreciates it that the referee has read the manuscript very carefully. References [1] F. Arnault, The Rabin-Monier theorem for Lucas pseudoprimes. Math. Comp. 66 (1997) 869–881 [2] R. Baillie, S. S. Wagstaff, Lucas pseudo-primes. Math. Comp. 35 (1980) 1391–1417 [3] R. Crandall, R. Pomerance, Prime numbers, a computational perspective. 2nd ed. Springer-Verlag, 2005. [4] J. 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Weisfeiler, One-dimensional affine group schemes. J. Algebra 66 (1980) 550–568 [15] H. C. Williams, On numbers analogous to the Carmichael numbers. Canad. Math. Bull. 20 (1977) 133–143 32 NORIYUKI SUWA Noriyuki Suwa Department of Mathematics, Faculty of Science and Engineerings Chuo University 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551 Japan e-mail address: [email protected] (Received August 6, 2010 ) (Revised December 25, 2010 )prime power pα. Corollary 2.11 has an importance in the proof of the main theorem. Lemma 2.10. Let p be a prime with (p, 2D) = 1. Then the sequence 0 Ker[U(D)(Z ) U(D)(F )] U(D)(Z ) U(D)(F ) 0 → p → p → p → p → is a splitting exact sequence, and Ker[U(D)(Z ) U(D)(F )] is isomorphic p → p to the additive group Zp. Moreover, 10 NORIYUKI SUWAν and 1 n ε(n) − ,p ε(p) ordpn 1 mod2 for some 2 2 − ≡ p n p n with ord (p ε(p)) < s. Y| 2 | − 2rν 1 (4) B˜ = 1+ − (m,p ε(p)). | sℓpsp| 2r 1 − p n − Y| α Example 3.5. Let p be a prime > 2 and n = p . Then B˜ℓpsp = B˜eℓpsp = B˜sℓpsp and its order is equal to p ε(p). The first case of (3) occurs when α is odd, and the second when α is− even. Remark 3.6. Let n be an odd integer > 1 and n ε(n) = 2sm with − (m, 2) = 1. Since (m, 2) = 1, ( I)m = I, and therefore I B˜ − − {± } ⊂ sℓpsp ⊂ B˜ B˜ . eℓpsp ⊂ ℓpsp Remark 3.7. Assume that D is a square. Put D = d2. Then an isomor- 1 phism of group schemes over Z[ ] 2Dν and ei 1 mod2 1+ min(s 1,si) ≡ − − for some i with s