a m 1 mod p a km 1 mod p k<s 1.6. n > 1 n 1= s m, (m, = 1 a n n a m 1 mod n a km 1 mod n k<sn a 1.7. n > 1 n 1= s m, (m, = 1 r n ν = min ord (p 1 (1 B

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1 Journal of the Institute of Science and Engineering. Chuo University Euler n > 1 p n p ord p n n n 1= s m (m B psp = {a (Z/nZ ; a n 1 =1}, B epsp = { ( a (Z/nZ ; a n 1 a }, = n B spsp = { a (Z/nZ ; a m =1 a km = 1 0 k<s } 1.1. n > 1 a n 1 1 mod n, (a, =1 a n p p a Fermat a p 1 1 mod p 1.. n > 1 a n a n 1 1 mod n n a 1.3. n > 1 a n 1 mod n, (a, =1 a n ( p > p a Euler a p 1 a mod p p 1.4. n > 1 a n a n 1 Euler mod n n a 1.5. n > 1 n 1= s m, m a m ±1 mod n, a km 1 mod n (1 k<s, (a, =1 a n p > p 1= s m, m, p a (a m 1(a m + 1(a m +1 (a s 1m +1=a sm 1 0 mod p BB 1

2 a m 1 mod p a km 1 mod p k<s 1.6. n > 1 n 1= s m, (m, = 1 a n n a m 1 mod n a km 1 mod n k<sn a 1.7. n > 1 n 1= s m, (m, = 1 r n ν = min ord (p 1 (1 B psp B epsp B spsp ( B psp (Z/nZ B psp (n 1,p 1 (3 B epsp (Z/nZ B epsp (a s = ν (n 1,p 1 (b s>ν ord (p 1 <s p ord p n (n 1,p 1 (c s>ν ord (p 1 <s ord p n n p 1 (n 1,p 1 (4 B spsp ( 1+ rν 1 r 1 (m, p p > n = p α B psp = B epsp = B spsp p n > 1 n 1 = s m, (m, = 1 m ( 1 m = 1 {±1} B spsp B epsp B psp n > 1 n B psp =(Z/nZ n Carmichael Carmichael n = p 1 p p r (p 1,p,...,p r n Carmichael r 3 n 1 p 1 1, p 1,...,p r 1 1.7( n Carmichael ϕ( = (n 1,p 1 ϕ( = 1 (p n 1 = (p (n 1,p 1 n p n 1 p 1

3 Euler r =, n = pq (p <q n 1=(p 1q +(q 1 (q 1 (n 1 (q 1 (p 1 p<q (1 Pomerance, Selfridge, Wagstaff Monier ([10, Th.3], [9, Th.9] ( Baillie, Wagstaff Monier ([,Th.1], [9,Th.1 ] (3 Monier ([9, Prop.3] (4 Monier ([9, Prop.1] ((3(4 Monier Ribenboim [1, Ch..VIII] B psp =#{a Z/nZ ; a n 1 =1,a 1}, B epsp =# { ( a (Z/nZ ; a n 1 a =,a 1 }, B spsp =# { a Z/nZ ; a m =1 a km = 1 k<s, a 1 } 1 (Z/nZ [], [7], [9], [10], [11], [14] Miller [7] Riemann 1.5 Rabin [11] Miller Miller [7] a n 1 1 mod n k<s (a km 1,=1 n n a Rabin [11] Miller 1.11 Korselt [6] Carmichael [3] Carmichel Alford, Granville, Pomerance [1].1. n > 1 d n 1 H = {a (Z/nZ ; a d 1 mod n} H (Z/nZ H (d, p 1 n = p e1 1 pe per r (p 1,p,...,p r i a i = a mod p ei i a (a 1,a,...,a r (Z/nZ (Z/p e1 1 Z (Z/p e Z (Z/p er r Z a d 1 mod n i a d 1 mod p ei i i a (Z/p ei i Z d i a (Z/p ei i Z (d, ϕ(p ei i d n 1 d p i (d, ϕ(p ei i = (d, (p i 1p ei 1 i =(d, p i 1 (Z/p ei i Z (d, p i 1 (Z/p ei i Z (d, p i 1.. n > 1 k 1 ν = min ord (p 1 a k 1 mod n a Z k ν 1 n p p 1 k+1 3

4 n = p e1 1 pe per r (p 1,p,...,p r a k 1 mod n (Z/p ei i Z i a k 1 mod p ei i i a (Z/p ei i Z k+1 (Z/p ei i Z k+1 k+1 ϕ(p ei i k+1 (p i 1.3. n > 1 ν = min ord (p 1 ord (n 1 ν ord (n 1 = ν ord (n 1 >ν p n ord (p 1=ν ord (p 1=ν ord p n 1 mod, ord p n 0 ord (p 1 = ν p 1+ ν mod ν+1, mod ord (p 1 >ν p 1 mod ν+1 ( n 1+ ord (p 1=ν ord p n ν mod ν+1.4. n > 1 ν = min ord (p 1 n p ord (p 1 ord (n 1 ord (n 1 = ν, ord (p 1 < ord (n 1 n p ord (n 1 >ν.5. n > 1 a ν = min ord (p 1 (1 a ν 1 1 mod n =1 ( a ν 1 1 mod n ord (n 1 >ν =1 ( a (3 a ν 1 1 mod n ord (n 1 = ν = 1 a ν 1 1 mod n Euler n p a p 1 =(a ν 1 p 1 ν 1 mod p p 4

5 Euler =1 a ν 1 1 mod n p 1 ν 0 mod ord (p 1 >ν, p 1 ν 1 mod ord (p 1 = ν 1 ord (p 1 >ν = p 1 ord (p 1 = ν e = ord (p 1=ν ord p n =( 1 e.3 e 0 e 1 mod ord (n 1 >ν, mod ord (n 1 = ν (1 a B epsp a n 1 ±1 mod n a n 1 (±1 = 1 mod n a B psp a B spsp. a m 1 mod n a km 1 mod n k ν 1 s = ν a n 1 = a ν 1m ±1 mod n.5 1 a = ν 1m 1 mod n 1 a ν 1m 1 mod n a B epsp s>ν.5 a n 1 = a s 1m =(a ν 1m s ν 1 mod n =1 a B epsp (.1 d = n 1 5

6 (3 C = {a Z/nZ ; a n 1 =1} C (Z/nZ.1 C (n 1,p 1 (a s = ν a C a ν 1m 1 mod n m.5 m m = = =1 n a B epsp (. a ν 1 1 mod n a Z.5 a = 1 a B epsp a B epsp {±1} C C B epsp (b(c s>ν. a n 1 = a s 1m 1 mod n a Z B epsp C a n 1 1 mod n p n ord (p 1 s =1a p 1 ±1 mod p p m p a p 1 (a p 1 m =(a s 1m p 1 s 1 mod p ord (p 1 <s n p ord p n a C =1 B epsp = C ord p n ord (p 1 <s= ord (n 1 n ( p a1 n = p e1 1 pe per r (p 1,p,...,p r p 1 = p = 1 a 1 p 1 a p mod p 1 a 1 a pe a p mod p e1 1 s 1 = ord (p 1 1 p 1 1 s1 p ( 1 1 a s 1 ( 1 a1 p1 1 = s 1 = 1 p 1 p 1 p 1 1 a 1 a 1 1 a s mod p e1 1 s 1 1 <s 1 a s mod p e1 n 1 1 a1 1 mod p e1 1 a a a 1 mod p e1 1,a 1 mod pe,..., a 1 mod per r a n 1 1 mod n = 1, =1,..., =1 p 1 p p r e 1 = 1 a C {±1} B epsp B epsp C (4 k 0 C k = {a (Z/nZ ; a km =1} C k (Z/nZ (.1 C k k m, p 1 k ν ( k m, p 1 = rk ( m, p 1 6

7 Euler k 0 B k = {a (Z/nZ ; a km = 1} m B k ø c k 1 mod n c B k ø a ca C k Bk. c k 1 mod n c k +1 ν k ν B k =ø B spsp B spsp =(1+1+ r + + r(ν 1 B spsp = C 0 B 0 B 1 B ν 1 ( m, p 1 = (1+ rν 1 r 1 ( m, p 1.8. n > 1 n 1= s m, (m, = 1 r n ν = min ord (p 1 B spsp (Z/nZ r =1 ν =1 (n 1,p 1 (m, p 1 B psp : C 0 B spsp (Z/nZ B spsp : C 0 B spsp : C 0 =1+1+ r + + (ν 1r r =1 ν =1.9. n > 1 n 1 = s m, (m, = 1 ν = min ord (p 1 C B spsp (Z/nZ C = C ν 1 B ν 1 C ν 1 C 0 B 0 B 1 B ν C = C ν 1 = 1+r(ν 1 (m, p n > 1 n = p e1 1 pe per r (p 1,p,...,p r n n 1= s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i 1 i r ord ϕ( = s i ord B psp = min(s, s i, 1+r(s 1 ord B epsp = min(s 1,s i 1+ min(s 1,s i ord C =1+r(ν 1 s = ν s>ν s i <s i e i s>ν s i <s e i i 7

8 B psp B epsp C r (m, m i Koblitz [5], n > 1 B psp = B epsp n n n p ord (p 1 < ord (n 1 n = p e1 1 pe per r (p 1,p,...,p r n 1= s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i.10 B psp 1 i r B epsp B psp = B epsp ord B psp = ord B epsp s>ν s i <s e i i ord B psp = min(s, s i > 1+ min(s 1,s i = ord B epsp B psp B epsp s >ν s i <s i e i ord B psp = min(s, s i, ord B epsp = min(s 1,s i B psp = B epsp min(s, s i = min(s 1,s i i s i <s e i n s = ν ord B psp = min(s, s i =rs, ord B epsp =1+ min(s 1,s i =1+r(s 1 B psp = B epsp rs =1+r(s 1 r =1 3.. n B epsp ϕ(/ B epsp (Z/nZ B epsp (Z/nZ n Carmichael B epsp B psp <ϕ(n Carmichael n B epsp < B psp 3.3. n > 1 s = ord (n 1, ν= min ord (p 1 C B spsp (Z/nZ C B epsp s = ν n n = p α q β (p, q ord (p 1 = ord (q 1, α β 1 mod n = p e1 1 pe per r (p 1,p,...,p r n 1= s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i.10 C B epsp 1 i r B epsp = C ord B epsp = ord C ord C =1+r(ν 1 8

9 Euler s = ν ord B epsp =1+r(ν 1 B epsp = C s >ν s i <s i e i ord B epsp = i min(s 1,s i ν min(s 1,s i B epsp = C min(s 1,s i =1+r(ν 1 r =1 s >ν s i <s e i i ord B epsp = 1+ i min(s 1,s i ν min(s 1,s i B epsp = C 1+ min(s 1,s i =1+r(ν 1 r =,s 1 = s = ν e 1 e mod e 1,e e 1 e 1 mod 3.4. n > 1 B epsp = B spsp n 3 mod 4 n n = p α q β (p, q p q 3 mod 4, α β 1 mod n = p e1 1 pe per r (p 1,p,...,p r n 1= s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i 1 i r C B spsp (Z/nZ B epsp = B spsp B epsp = C C = B spsp 3.3 B epsp = C s = ν r =1 r =, s 1 = s = ν, e 1 e 1 mod.8 C = B spsp r =1 ν =1 B epsp = B spsp s =1 r =1 r =,s 1 = s =1,e 1 e 1 mod 3.5. n > 1 (1 B epsp = ϕ(/ n Carmichael n p ord (p 1 < ord (n 1 ( B epsp = ϕ(/4 n = pq (p, q q =p 1 n p 1 p p 3 (p 1,p,p 3 ord (p 1 1 = ord (p 1 = ord (p 3 1 Carmaichael n Carmichael 9

10 p ord (p 1 = ord (n 1 q ord (q 1 < ord (n 1 n = p e1 1 pe per r (p 1,p,...,p r n 1 = s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i ord ϕ( = s i 1 i r (1 B epsp = ϕ(/ n i m i m r s = ν.10 ord B epsp =1+r(s 1 1+r(s 1 = 1+ s i (s i s +1= i s i s +1 1 r =,s 1 = s = s.3 s>ν s = ν s >νn.10 ord B epsp = 1+ min(s 1,s i 1+ min(s 1,s i = 1+ s i min(s 1,s i = s i i s i s 1 i m i m n Carmichael n Carmichael i s i <s i (p i 1 n 1 s >ν n.10 B epsp = 1 r (n 1,p i 1 = 1 r (p i 1 = ϕ( ( B epsp = ϕ(/4 n i m i m r s = ν.10 ord B epsp =1+r(s 1 1+r(s 1 = + s i (s i s +1=3 10

11 Euler i s i s +1 1 r =,s 1 = s, s = s +1 r =3,s 1 = s = s 3 = s (a r =,s 1 = s, s = s+1 m 1 m (p 1 1 (n 1 n 1 =(p 1 1p +(p 1 (p 1 1 (p 1m m (p 1 (n 1 (n 1 = (p 1p 1 +(p 1 1 (p 1 (p 1 1 p 1=(p 1 1 (b r =3, s 1 = s = s 3 = s i (m, m i =m i n Carmichael s >νn.10 ord B epsp = 1+ min(s 1,s i min(s 1,s i = + min(s 1,s i = s i s i s i = s i j i s 1 s i i m i m n Carmichael n = p 1 p p 1=(p 1 1 p 1 1= s1 m 1, p 1= s1+1 m 1,n 1= s1 m 1 (3 + s1+1 m 1 s = s 1 +1,m = m 1, s = s 1,m 1 m.10 ord B epsp =1+(s 1 = +(s 1 + s = + ord ϕ( n p 1 p p 3 (s 1 = s = s 3 Carmaichael.3 s = ν.10 ord B epsp =1+3(s 1 = +(s 1 + s + s 3 = + ord ϕ( n Carmichael s 1 = s i s i <ss >ν n.10 ord B epsp = 1+ min(s 1,s i = 1+(s B epsp = ϕ(/ (1 B epsp = ϕ(/ n< = = = = = = ( B epsp = ϕ(/4 n< s i = + i= s i = + ord ϕ( 11

12 15 = = = = = = = = = = = = = = = = = = = = n (1 n 9 B spsp ϕ(/4 ( B spsp = ϕ(/4 n = pq (p, q p 3 mod 4 q =p 1 n p 1 p p 3 (p 1,p,p 3 p 1 p p 3 3 mod 4 Carmaichael n = p e1 1 pe per r (p 1,p,...,p r n 1= s m, (m, = 1 i p i 1= si m i, (m i, = 1 ν = min s i 1 i r 3. B spsp B epsp ϕ(/ (a B epsp = ϕ(/ 3.5 n Carmichael i s i <s C B spsp (Z/nZ.10 C B epsp ord B epsp ord C ( = 1+ s i {1+r(ν 1} = + (s i ν +1 + (s i ν +1 n Carmichael r 3 i s i ν (s i ν+1 (s i ν +1=1 r =3 s 1 = s = s 3 =1.3 s =1 s>ν 1

13 Euler B spsp C B epsp /4 =ϕ(/8 (b B epsp = ϕ(/3 n =9 B spsp = ϕ(/3 (c B epsp = ϕ(/4 3.5 r =, p = p 1 1, n Carmichael r =3, s 1 = s = s 3, n Carmichael i s i = s j i s j <s 3.4 B spsp = B epsp s =1 r =1 r =,s 1 = s =1,e 1 e 1 mod B spsp = ϕ(/4 r =,s=1,p =p 1 1 n Carmichael r =3,s 1 = s = s 3 =1 p =p 1 1 s = s B spsp = ϕ(/4 n< = = = = = = = = = n > 1 n 1= s m,(m, = 1 ν = min ord (p 1 (1 B epsp = {±1} n 3 n 3 mod 4 n p (m, p 1 = 1 n = p α q β (p, q p q 3 mod 4, (m, p 1 = (m, q 1 = 1, α β 1 mod ( B spsp = {±1} ν =1 n p (m, p 1 = Solovay, Strassen [14] 3.7(1 Monier Rabin ([9, Prop.1], [11, Th.1] Monier [9] Prop.3 3.5(1 Prop.1 3.7( Rabin [11] Th.1 n p 1 p p 3 (p 1,p,p 3 p 1 p p 3 3 mod 4 Carmaichael B spsp = ϕ(/4 Monier [9] Th n 3 mod 4 B epsp = B spsp Malm [8] [4] 3 [5] 5 1 [1] 8 9 Euler Carmichael Euler [13] 13

14 [1] W. R. Alford, A. Granville, C. Pomerance, There are infinitely many Carmichael numbers. Ann. of Math. 140 ( [] R. Baillie, S. S. Wagstaff Jr., Lucas pseudo-primes. Math. Comp. 35 ( [3] R. D. Carmichael, On composite numbers P which satisfy the Fermat congruence a P 1 1 (mod P. Amer. Math. Monthly 19 (191 7 [4] UBASIC (1994 [5] N. Koblitz, A course in number theory and cryptography, nd ed. Springer-Verlag (1994/ ( (1996 [6] A. Korselt, Problème chinois. L Intermédiaire des Mathématiciens 6 ( [7] G. L. Miller, Riemann s hypothesis and a test for primality. J. Comput. and System Sci. 13 ( [8] D. E. G. Malm, On Monte-Carlo primality tests. Notices Amer. Math. Soc. 4 ( [9] L. Monier, Evaluation and comparison of two efficient probabilistic primality testingalgorithm. Theoret. Comput. Sci. 1 ( [10] C. Pomerance, J. L. Selfridge, S. S. Wagstaff Jr., The pseudoprimes to Math. Comp. 35 ( [11] M. O. Rabin, Probabilistic algorithm for testing primality. J. Number Theory 1 ( [1] P. Ribenboim, The little book of bigprimes, Springer-Verlag(1991/ ( (001 [13], 00 [14] R. Solovay, V. Strassen, A fast Monte-Carlo test for primality. SIAM J. Comput. 6 ( [15] H. C. Williams, Primality testingon a computer. Ars Comb. 5 (

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