aa r X i v : . [ m a t h . N T ] M a y THE ARITHMETIC OF CARMICHAEL QUOTIENTS
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Abstract.
Carmichael quotients for an integer m ≥ λ ( m ). Various properties of these new quotients are investi-gated, such as basic arithmetic properties, sequences derived fromCarmichael quotients, Carmichael-Wieferich numbers, and so on.Finally, we link Carmichael quotients to perfect nonlinear func-tions. Introduction
Let p be a prime and a an integer not divisible by p , by Fermat’slittle theorem, the Fermat quotient of p with base a is defined as follows Q p ( a ) = a p − − p . Moreover, if Q p ( a ) ≡ p ), then we call p a Wieferich prime withbase a .This quotient has been extensively studied from various aspects be-cause of its numerous applications in number theory and computerscience; see, for example, [7, 8, 9, 11, 16, 17]. A first comprehensivestudy of Fermat quotient was published in 1905 by Lerch [12], whichwas based on the viewpoint of arithmetic. More arithmetic propertieswere investigated in [3].In [4], the authors generalized the definition of Fermat quotient byusing Euler’s theorem. Let m ≥ a be relatively prime integers,the Euler quotient of m with base a is defined as follows Q m ( a ) = a ϕ ( m ) − m , where ϕ is Euler’s totient function. Moreover, if Q m ( a ) ≡ m ),then we call m a Wieferich number with base a . They also undertooka very careful study of Euler quotients. Mathematics Subject Classification.
Key words and phrases.
Carmichael function, Carmichael quotient, Carmichael-Wieferich number, perfect nonlinear function.
In fact, there are some other generalizations of Fermat quotients,see [1, 18, 19]. Especially, in [1] the author introduced a quotient like( a e − /m , where gcd( a, m ) = 1 and e is the multiplicative order of a modulo m .In this paper, we introduce a different generalization of Fermat quo-tient by using Carmichael function and study its arithmetic properties.For a positive integer m , the Carmichael function λ ( m ) is defined tobe the exponent of the multiplicative group ( Z /m Z ) ∗ . More explicitly, λ (1) = 1; for a prime power p r we define λ ( p r ) = (cid:26) p r − ( p −
1) if p ≥ r ≤ , r − if p = 2 and r ≥ λ ( m ) = lcm( λ ( p r ) , λ ( p r ) , · · · , λ ( p r k k )) , where, as usual, “lcm” means the least common multiple, and m = p r p r · · · p r k k is the prime factorization of m .For every positive integer m , we have λ ( m ) | ϕ ( m ), and λ ( m ) = ϕ ( m )if and only if m ∈ { , , , p k , p k } , where p is an odd prime and k ≥ m | n , we have λ ( m ) | λ ( n ). Definition 1.1.
Let m ≥ a be relatively prime integers. Thequotient C m ( a ) = a λ ( m ) − m is called the Carmichael quotient of m with base a . Moreover, if C m ( a ) ≡ m ), we call m a Carmichael-Wieferich number withbase a .We want to indicate that the term “Carmichael quotient” was intro-duced in [2] to denote a different quotient, and we think that there isno much danger of confusion.We extend many known results about Fermat quotients or Euler quo-tients to Carmichael quotients by using the same techniques, such asbasic arithmetic properties with special emphasis on congruences, theleast periods of sequences derived from Carmichael quotient, Carmichael-Wieferich numbers. Finally, we link Carmichael quotients to perfectnonlinear functions.2. Arithmetic of Carmichael Quotients
In what follows, we fix m ≥ HE ARITHMETIC OF CARMICHAEL QUOTIENTS 3
For any integer a with gcd( a, m ) = 1, we have C m ( a ) | Q m ( a ). In par-ticular, C m ( a ) = Q m ( a ) when m is an odd prime power. Furthermore,it is straightforward to prove that they have the following relation. Proposition 2.1.
For any integer a with gcd( a, m ) = 1 , we have Q m ( a ) ≡ ϕ ( m ) λ ( m ) · C m ( a ) (mod m ) . Proof.
Since λ ( m ) | ϕ ( m ), we derive Q m ( a ) = ( a λ ( m ) ) ϕ ( m ) /λ ( m ) − m = ( a λ ( m ) − (cid:0) a λ ( m ) + · · · + ( a λ ( m ) ) ϕ ( m ) /λ ( m ) − (cid:1) m ≡ ϕ ( m ) λ ( m ) C m ( a ) (mod m ) . (cid:3) Now we state two fundamental congruences for Carmichael quotients,which are crucial for further study.
Proposition 2.2. (1)
If a and b are integers with gcd( ab, m ) = 1 , thenwe have C m ( ab ) ≡ C m ( a ) + C m ( b ) (mod m ) . (2) If a, k are integers with gcd( a, m ) = 1 , and α is a positive integer,then we have C m ( a + km α ) ≡ C m ( a ) + kλ ( m ) a m α − (mod m α ) . Proof. (1) We only need to notice that C m ( ab ) = a λ ( m ) b λ ( m ) − m = ( a λ ( m ) − b λ ( m ) −
1) + ( a λ ( m ) −
1) + ( b λ ( m ) − m . (2) Using the binomial expansion, it is easy to see that C m ( a + km α ) ≡ a λ ( m ) + λ ( m ) a λ ( m ) − km α − m (mod m α ) , which implies the desired congruence. (cid:3) The following two corollaries concern some short sums of Carmichaelquotients.
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Corollary 2.3. If m ≥ , for any integer a with gcd( a, m ) = 1 , wehave m − X k =0 C m ( a + km ) ≡ m ) . Proof.
First applying Proposition 2.2 (2) and then noticing that λ ( m )is even when m ≥
3, we obtain m − X k =0 C m ( a + km ) ≡ λ ( m ) a · m ( m − ≡ m ) . (cid:3) Corollary 2.4. If m ≥ , for any integer a with gcd( a, m ) = 1 , wehave m X a =1gcd( a,m )=1 C m ( a ) ≡ m ) . Proof.
Notice that m X a =1gcd( a,m )=1 C m ( a ) = m X a =1gcd( a,m )=1 m − X k =0 C m ( a + km ) . Then, the desired result follows from Corollary 2.3. (cid:3)
We want to remark that the results in Corollaries 2.3 and 2.4 are nottrue when m = 2.The next proposition concerns some relationships between various C m ( a ) with fixed base a and different moduli. Proposition 2.5. (1) If gcd( a, mn ) = 1 , then C m ( a ) | nC mn ( a ) . (2) If gcd( a, mn ) = gcd( m, n ) = 1 , then C mn ( a ) ≡ λ ( n ) n · gcd( λ ( m ) , λ ( n )) C m ( a ) (mod m ) . (3) Assume that gcd( a, mn ) = gcd( m, n ) = 1 , and let X and Y be twointegers satisfying m X + n Y = 1 . Then C mn ( a ) ≡ nλ ( n )gcd( λ ( m ) , λ ( n )) Y C m ( a )+ mλ ( m )gcd( λ ( m ) , λ ( n )) XC n ( a ) (mod mn ) . HE ARITHMETIC OF CARMICHAEL QUOTIENTS 5
Proof. (2) Under the assumption, noticing that λ ( mn ) = λ ( m ) λ ( n )gcd( λ ( m ) ,λ ( n )) ,we have C mn ( a ) = a λ ( m ) λ ( n )gcd( λ ( m ) ,λ ( n )) − mn = ( a λ ( m ) ) λ ( n )gcd( λ ( m ) ,λ ( n )) − mn ≡ λ ( n )( a λ ( m ) − mn · gcd( λ ( m ) ,λ ( n )) (mod m ) . (3) It suffices to show that the equality is true for modulo m andmodulo n respectively. But this follows directly from (2). (cid:3) For any integer a with gcd( a, m ) = 1, we denote h a i as the subgroupof ( Z /m Z ) ∗ generated by a , and we let ord m a be the multiplicativeorder of a modulo m . The following expression is so-called Lerch’sexpression [13]. Proposition 2.6. If gcd( a, m ) = 1 and assume n = ord m a , then C m ( a ) ≡ λ ( m ) n m X r =1 r ∈h a i ar j arm k (mod m ) , where ⌊ x ⌋ denotes the greatest integer ≤ x .Proof. For each 1 ≤ r ≤ m with r ∈ h a i , we write ar ≡ c r (mod m ),with 1 ≤ c r ≤ m . Notice that when r runs through all elements with1 ≤ r ≤ m and r ∈ h a i , so does c r . Let P denote the product of allsuch integers c r . If the products and sums below are understood to betaken over all r with 1 ≤ r ≤ m and r ∈ h a i , we have P λ ( m ) n = Y c λ ( m ) n r = Y (cid:16) ar − m j arm k(cid:17) λ ( m ) n = a λ ( m ) P λ ( m ) n Y (cid:16) − mar j arm k(cid:17) λ ( m ) n . So1 = a λ ( m ) Y (cid:16) − mar j arm k(cid:17) λ ( m ) n ≡ a λ ( m ) (cid:18) − m X ar j arm k(cid:19) λ ( m ) n (mod m ) . Then we get a λ ( m ) − ≡ a λ ( m ) mλ ( m ) n m X r =1 r ∈h a i ar j arm k (mod m ) , which implies the desired congruence. (cid:3) In the last part of this section, we describe the decomposition ofCarmichael quotients in the dependence of the prime factorization ofthe modulus. Further we investigate Carmichael quotients for primepower moduli.
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Proposition 2.7.
Let m = p r · · · p r k k be the prime factorization of m , and let a be an integer with gcd( a, m ) = 1 . For ≤ i ≤ k , let d i = λ ( m ) /λ ( p r i i ) , m i = m/p r i i and m ′ i ∈ Z such that m i m ′ i ≡ p r i i ) . Then C m ( a ) ≡ k X i =1 m i m ′ i d i C p rii ( a ) (mod m ) . Proof.
It suffices to prove for each 1 ≤ j ≤ k , C m ( a ) ≡ k X i =1 m i m ′ i d i C p rii ( a ) (mod p r j j ) , that is C m ( a ) ≡ m j m ′ j d j C p rjj ( a ) (mod p r j j ) . Since we have C m ( a ) = a λ ( p rjj ) d j − m ≡ d j ( a λ ( p rjj ) − m ≡ m j m ′ j d j C p rjj ( a ) (mod p r j j ) , the result follows. (cid:3) Proposition 2.8.
Let p be an odd prime and gcd( a, p ) = 1 . For anytwo integers i and j with ≤ i ≤ j , we have C p j ( a ) ≡ C p i ( a ) (mod p i ) . Besides, for ≤ i ≤ j and gcd( a,
2) = 1 , we have C j ( a ) ≡ C i ( a ) (mod 2 i − ) . Proof.
Notice that C p i ( a ) = Q p i ( a ) if p is an odd prime. By [4, Propo-sition 4.1], for any integer k ≥
1, we have C p k +1 ( a ) ≡ C p k ( a ) (mod p k ) . Then the first formula follows.Since for r ≥
3, we have C r +1 ( a ) − C r ( a ) ≡ a r − − C r ( a ) (mod 2 r ) ≡ r − ) , we get the second formula. (cid:3) The following corollary, about the relation between Carmichael quo-tients and Fermat quotients, can be obtained directly from the abovetwo propositions.
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Corollary 2.9.
Suppose that p is an odd prime factor of m , and p α is the largest power of p dividing m . Let d = λ ( m ) λ ( p α ) , m = m/p α , and m ′ ∈ Z such that m m ′ ≡ p α ) . Then for any integer a with gcd( a, m ) = 1 , we have C m ( a ) ≡ m m ′ d Q p ( a ) (mod p ) . Sequences derived from Carmichael quotients
In this section, we will define two periodic sequences by Carmichaelquotients and determine their least (positive) periods following themethod in the proof of [10, Proposition 2.1].As usual, for a periodic sequence { s n } ∞ n =1 , a positive integer j iscalled its period if s n + j = s n for any n ≥
1; if further j is the smallestpositive integer endowed with such property, we call j the least period of { s n } .Let m = p r · · · p r k k be the prime factorization of the integer m ( m ≥ ≤ i ≤ k , put m i = m/p r i i , and let w i be the integerdefined by p w i i = gcd( λ ( m ) /λ ( p r i i ) , p r i i ), here note that 0 ≤ w i ≤ r i .Now, we want to define a sequence { a n } following the manner in [10].First, for any integer n and any 1 ≤ i ≤ k , if p i | n , set C p rii ( n ) = 0.Then, for every integer n ≥
1, by Proposition 2.7, a n is defined as theunique integer with a n ≡ k X i =1 m i m ′ i λ ( m ) λ ( p r i i ) C p rii ( n ) (mod m ) , ≤ a n ≤ m − , where m ′ i ∈ Z is such that m i m ′ i ≡ p r i i ) for each 1 ≤ i ≤ k . So,if gcd( n, m ) = 1, we have a n ≡ C m ( n ) (mod m ).By Proposition 2.2 (2), m is a period of { a n } . We denote its leastperiod by T . For each 1 ≤ i ≤ k , let T i be the least period of thesequence { a n mod p r i i } . Obviously, we have T = lcm( T , · · · , T k ) . Thus, in order to determine T , it suffices to compute T i for each 1 ≤ i ≤ k .For every 1 ≤ i ≤ k , we have(3.1) a n ≡ λ ( m ) m i λ ( p r i i ) C p rii ( n ) (mod p r i i ) . So, T i equals to the least period of { C p rii ( n ) mod p r i − w i i } . Here, wealso denote T i as the least period of the sequence { C p rii ( n ) mod p r i − w i i } without confusion. In the sequel, we will calculate T i case by case forany fixed 1 ≤ i ≤ k . MIN SHA
Lemma 3.1. If w i = r i , then T i = 1 .Proof. Since in this case we have C p rii ( n ) ≡ p r i − w i i ) for all n ≥ (cid:3) Lemma 3.2. If p i > and w i < r i , then T i = p r i − w i +1 i .Proof. Combining Proposition 2.2 (2) with Proposition 2.8, for integers n and ℓ with gcd( n, p i ) = 1, we have C p rii ( n + ℓp r i − w i i ) ≡ C p ri − wii ( n + ℓp r i − w i i ) ≡ C p ri − wii ( n ) + ℓn − ( p i − p r i − w i − i ≡ C p rii ( n ) + ℓn − ( p i − p r i − w i − i (mod p r i − w i i ) . Thus, T i = p r i − w i +1 i . (cid:3) Now, it remains to consider the case p i = 2. Lemma 3.3. If p i = 2 and w i = 0 , then T i = r i = 1 , r i = 2 , r i +2 r i ≥ Proof.
Notice that for each n with gcd( n,
2) = 1, by Proposition 2.2(2) we have C ri ( n + ℓ · r i ) ≡ C ri ( n ) + ℓn − λ (2 r i ) (mod 2 r i ) . Then, the result follows easily. (cid:3)
Lemma 3.4.
For r ≥ , the least period of the sequence { C r +1 ( n )mod 2 r } is r +2 .Proof. For r ≥ n,
2) = 1, we have C r +1 ( n ) = n r − +12 C r ( n ).Then using Proposition 2.2 (2), we deduce that C r +1 ( n + ℓ · r ) − C r +1 ( n ) = n r − +12 ( C r ( n + ℓ · r ) − C r ( n )) ≡ n r − +12 · ℓn − r − (mod 2 r ) , which implies the desired result by noticing that n r − ≡ r )and then n r − +12 is odd. (cid:3) Lemma 3.5. If p i = 2 and ≤ r i − w i < r i , then T i = 2 r i − w i +2 .Proof. By Proposition 2.8, for gcd( n,
2) = 1, we have C ri ( n ) ≡ C ri − wi +1 ( n ) (mod 2 r i − w i ) . Then, the result follows directly from Lemma 3.4. (cid:3)
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Lemma 3.6. If p i = 2 , r i ≥ and ≤ r i − w i ≤ , then T i = 2 r i − w i +2 .Proof. From Proposition 2.8, for gcd( n,
2) = 1, we have C ri ( n ) ≡ C ( n ) (mod 2 ) . So, T i equals to the least period of the sequence { C ( n ) mod 2 r i − w i } .By Proposition 2.2 (2), we have C ( n + ℓ · ) ≡ C ( n ) + 2 ℓn − (mod 2 ) , which implies the desired result. In fact, one can also verify this lemmaby direct calculations. (cid:3) Lemma 3.7. If p i = 2 , r i = 2 and w i = 1 , then T i = 1 . We summarize the above results in the following proposition.
Proposition 3.8.
For each ≤ i ≤ k , if p i is an odd prime, then T i = (cid:26) w i = r i ,p r i − w i +1 i w i < r i ; otherwise if p i = 2 , then T i = w i = r i , r i = 1 , w i = 0 , r i = 2 , w i = 0 , r i = 2 , w i = 1 , r i − w i +2 r i ≥ , w i < r i . In particular, the least period of { a n } is T = T T · · · T k . When m = p r with p an odd prime and r ≥
1, we have T = p r +1 ,which is consistent with [10, Proposition 2.1]. If m = 2 r with r ≥ T = 2 r +2 ; but by [10, Proposition 2.1], the least period of thesequence defined there by Euler quotient is 2 r +1 .Finally, we want to define a new sequence { b n } , which is much simplerbut has the same least period as { a n } .For an integer n ≥ n, m ) = 1, b n is defined to be theunique integer with b n ≡ C m ( n ) (mod m ) , ≤ b n ≤ m − b n = 0 , if gcd( n, m ) = 1 . Since b n also satisfies (3.1) for any integer n with gcd( n, m ) = 1, theleast period of { b n } equals to that of { a n } . Proposition 3.9.
The sequence { b n } has the same least period as { a n } . Carmichael-Wieferich Numbers
In this section, except for extending some results in [4], we studyCarmichael-Wieferich numbers from more aspects, especially Proposi-tion 4.5.First, we want to deduce some basic facts for Carmichael-Wieferichnumbers.
Proposition 4.1. If m ≥ and ≤ a ≤ m with gcd( a, m ) = 1 , then m cannot be a Carmichael-Wieferich number with bases both a and m − a .Proof. Notice that λ ( m ) is even when m ≥
3. By Proposition 2.2 (2),we have C m ( m − a ) ≡ C m ( a ) − λ ( m ) a (mod m ) . Then, the desired result comes from λ ( m ) < m . (cid:3) Corollary 4.2. If m ≥ , define the set S m = { a : 1 ≤ a ≤ m, gcd( a, m ) =1 , m is a Carmichael-Wieferich number with base a } . Then | S m | ≤ ϕ ( m ) / . By Proposition 2.2 (2), for any gcd( b, m ) = 1, there exists 1 ≤ a ≤ m with b ≡ a (mod m ), such that C m ( b ) ≡ C m ( a ) (mod m ) . Hence, if we want to determine with which base m can be a Carmichael-Wieferich number, we only need to consider 1 ≤ a ≤ m .Assume that m has the prime factorization m = p r · · · p r k k . In [4,Proposition 4.4] the authors have used the Euler quotient Q m to define ahomomorphism from ( Z /m Z ) ∗ to ( Z /m Z , +), whose image is d Z /m Z ,where(4.1) d = k Y i =1 d i and d i = (cid:26) gcd( p r i i , ϕ ( m ) /ϕ ( p r i i )) if p i = 2 and r i ≥ , gcd( p r i i , ϕ ( m ) /ϕ ( p r i i )) otherwise . Here, we can do similar things using the Carmichael quotient and ap-plying the same strategy as in [4].By Proposition 2.2, the Carmichael quotient C m ( x ) induces a homo-morphism φ m : ( Z /m Z ) ∗ → ( Z /m Z , +) , x C m ( x ) . Proposition 4.3.
Let m = p r · · · p r k k be the prime factorization of m ≥ . For ≤ i ≤ k , put d ′ i = (cid:26) gcd( p r i i , λ ( m ) /λ ( p r i i )) if p i = 2 and r i = 2 , gcd( p r i i , λ ( m ) /λ ( p r i i )) otherwise . HE ARITHMETIC OF CARMICHAEL QUOTIENTS 11
Let d ′ = Q ki =1 d ′ i . Then the image of the homomorphism φ m is d ′ Z /m Z .Proof. We show the desired result case by case.(I) First we prove the result for the case k = 1, that is m = p r , where p is a prime and r is a positive integer.Suppose that p = 2. If r = 2, then C m (3) = 2, and for any positiveinteger n we have C m (2 n + 1) = n ( n + 1), which is even, so the imageof φ m is 2 Z /m Z . On the other hand, if r = 1 or r ≥
3, since C (3) = 1and C (3) = 1, by using Proposition 2.8 we see that C m (3) is an oddinteger, so the image of φ m is Z /m Z .Now, assume that p >
2. Note that C p ( p + 1) ≡ − p ),by Proposition 2.8 we have C m ( p + 1) ≡ − p ), which impliesthat p ∤ C m ( p + 1). Thus, there exists a positive integer n such that nC m ( p + 1) ≡ m ). Then, by Proposition 2.2 (1) we deducethat C m (( p + 1) n ) ≡ m ). So, the image of φ m is Z /m Z .(II) To complete the proof, we prove the result when k ≥ m i = m/p r i i and n i = λ ( m ) /λ ( p r i i ) for each1 ≤ i ≤ k , and then let m ′ i be an integer such that m i m ′ i ≡ p r i i ).By Proposition 2.7, we have(4.2) C m ( a ) ≡ k X i =1 m i m ′ i n i C p rii ( a ) (mod m ) . So, for each 1 ≤ i ≤ k , C m ( a ) ≡ m i m ′ i n i C p rii ( a ) (mod p r i i ). If p i = 2and r i = 2, note that for any odd integer a > C ( a ) is even, thenwe see that d ′ i | n i C p rii ( a ), and thus d ′ i | C m ( a ). Otherwise if p i > r i = 2, then d ′ i | n i , and so d ′ i | C m ( a ). Hence, we have d ′ | C m ( a ) forany integer a coprime to m .Let b = gcd( m, m m ′ n , . . . , m k m ′ k n k ). Then, there exist integers X , . . . , X k such that(4.3) b ≡ k X i =1 m i m ′ i n i X i (mod m ) . If we denote b i = gcd( p r i i , m i m ′ i n i ) for each 1 ≤ i ≤ k , then b = Q ki =1 b i ,here we remark that b i = gcd( p r i i , n i ). It is easy to see that for each1 ≤ i ≤ k , if p i > r i = 2, we have d ′ i = b i . Further, when p i = 2and r i = 2, d ′ i = 2 b i if 8 ∤ λ (2 p . . . p k ), and d ′ i = b i otherwise.We now have three cases for m :(i) There exists 1 ≤ j ≤ k such that p j = 2 , r j = 2 and8 ∤ λ (2 p . . . p k ) . (ii) There exists 1 ≤ j ≤ k such that p j = 2 , r j = 2 and8 | λ (2 p . . . p k ) . (iii) All the other cases.Clearly, in Cases (ii) and (iii) we have d ′ = b , and in Case (i) d ′ = 2 b .According to (I), there exist integers a i with p i ∤ a i for 1 ≤ i ≤ k defined by C p rii ( a i ) ≡ X i in Case (i) ,X i in Case (iii) , (mod p r i i ) X i in Case (ii) and i = j, i = j. By the Chinese Remainder Theorem, we can choose a positive integer a such that a ≡ a i (mod p r i i ). So, by Proposition 2.2 (2) we have C p rii ( a ) ≡ C p rii ( a i ) (mod p r i i ). Then, combining with (4.3) and therelation between b and d ′ , we obtain m i m ′ i n i C p rii ( a ) ≡ d ′ (mod p r i i ) foreach 1 ≤ i ≤ k in all the three cases. Finally, using (4.2) we have C m ( a ) ≡ d ′ (mod m ), which completes the proof. (cid:3) Comparing (4.1) with Proposition 4.3, we have d ′ | d . Moreover, byProposition 2.1 we get ϕ ( m ) λ ( m ) d ′ Z /m Z = d Z /m Z , which implies that gcd( ϕ ( m ) λ ( m ) d ′ , m ) = d .In Proposition 4.3, if choosing m = 2 r with r ≥
3, we have d = 2and d ′ = 1; while choosing m = 2 r p r with r ≥ p ≡ d = 4 and d ′ = 1. Hence, compared with[4, Proposition 4.4], the homomorphism φ m can be surjective in morecases.For any integer m ≥
2, we define the set T m = { a :1 ≤ a ≤ m , gcd( a, m ) = 1 ,m is a Carmichael-Wieferich number with base a } . Actually, T m is the kernel of the homomorphism φ m , then the followingresult follows directly from Proposition 4.3. Corollary 4.4.
We have | T m | = d ′ ϕ ( m ) , where d ′ is defined in Propo-sition 4.3. Corollary 4.4 shows that any integer m ≥ HE ARITHMETIC OF CARMICHAEL QUOTIENTS 13
Proposition 4.5.
We have lim m →∞ | T m | ϕ ( m ) = 0 . Proof.
Denote by d ( m ) the parameter d in (4.1). By Corollary 4.4, weknow that | T m | ϕ ( m ) ≤ d ( m ) m . So, it suffices to prove that lim m →∞ d ( m ) m = 0.For primes p , we havelim p →∞ d ( p ) p = lim p →∞ p = 0 . So lim inf m →∞ d ( m ) m = 0 . Suppose that lim sup m →∞ d ( m ) m = 0 . Then there exists a subsequence { d ( n i ) n i } such that lim i →∞ d ( n i ) n i = lim sup m →∞ d ( m ) m = 0 . For an integer m ≥
2, let m = p r · · · p r k k be its prime factorization.Put α m = max { r , · · · , r k } . Here we use the notation in (4.1). For each1 ≤ j ≤ k , we have d ( m ) m ≤ d j /p r j j . In particular, if p j is the largestprime factor of m , then d ( m ) m ≤ /p r j j .For each i , let p i be the largest prime factor of n i , we abbreviate α n i to α i . Since d ( n i ) n i ≤ p i for each i and lim i →∞ d ( n i ) n i = 0, there mustexist an integer q such that p i < q for all i . Put β = 2 Q ≤ p 2) = (cid:26) ord ( a − − a ≡ , ord ( a + 1) − a ≡ . Then, we can state an analogue of [4, Proposition 5.4]. For the conve-nience of the reader, we reproduce the proof. Proposition 4.6. Let gcd( a, m ) = 1 , and m = p r · · · p r k k be the primefactorization of m ≥ . Fix an integer j with ≤ j ≤ k , let p = p j and r = r j . If p = 2 or r ≤ , put n = (cid:26) if ord p lcm ( p − , · · · , p k − ≤ r − , ord p lcm ( p − , · · · , p k − − r + 1 otherwise ; otherwise if p = 2 and r > , put n = (cid:26) if ord p lcm ( p − , · · · , p k − ≤ r − , ord p lcm ( p − , · · · , p k − − r + 2 otherwise . Moreover, put e ( m, p ) = (cid:26) n if p = 2 or r ≤ ,n − otherwise . Then we have ord p C m ( a ) = e ( m, p ) + σ ( a, p ) . Proof. Notice that λ ( m ) = p n λ ( p r ) X , where X is an integer with p ∤ X .Put b = a p n λ ( p r ) . Then, since a λ ( m ) − b X − b − X − X i =0 b i ,b ≡ p ) and P X − i =0 b i ≡ X p ), we obtainord p ( a λ ( m ) − 1) = ord p ( b − 1) = ord p ( a p n λ ( p r ) − . Thus, if p is an odd prime, by using [4, Lemma 5.1] we haveord p ( a λ ( m ) − 1) = ord p (( a p − ) p n + r − − 1) = ord p ( a p − − 1) + n + r − , which implies that ord p C m ( a ) = e ( m, p ) + σ ( a, p ) . Similarly, applying [4, Lemmas 5.1 and 5.3], one can verify the remain-ing case p = 2 by noticing that m ≥ (cid:3) The next proposition, a criterion for a number m being a Carmichael-Wieferich number, follows directly from Proposition 4.6. HE ARITHMETIC OF CARMICHAEL QUOTIENTS 15 Proposition 4.7. Let gcd( a, m ) = 1 , and m = p r · · · p r k k be the primefactorization of m ≥ . Then the following statements are equivalent: (1) m is a Carmichael-Wieferich number with base a , (2) e ( m, p j ) + σ ( a, p j ) ≥ r j , for any ≤ j ≤ k . Although it is known that Wieferich primes exist for many differentbases (see [15]), the following problem is still open. Whether Wieferich primes exist for all bases? Proposition 4.8. For a non-zero integer a , if there exists a Carmichael-Wieferich number m with base a and m has an odd prime factor, thenthere exists a Wieferich prime with base a .Proof. Let m = p r · · · p r k k be the prime factorization of m with p
1. Notice that p k is an odd prime, so p k is a Wieferichprime with base a . (cid:3) Finally, we want to remark that a Carmichael-Wieferich number m with base a is also a Wieferich number with base a , but the converseis not true. Example 4.9. From Table 1 of [15], 3 and 7 are two Wieferich primeswith base 19. It is straightforward to see that 2 is not a Wieferich primewith base 19. By [4, Theorem 5.5], m = 2 · · m is not a Carmichael-Wieferichnumber with base 19.5. Involving perfect nonlinear function Let ( A, +) and ( B, +) be two additive abelian groups, and denoteby ¯ A the set of non-identity elements of A . When | A | is a multiple of | B | , we can consider the following definition; see [5] for more details. Definition 5.1. Let f : A → B be a function from A to B . Then f is called perfect nonlinear if for every ( a, b ) ∈ ¯ A × B , |{ x ∈ A : f ( x + a ) − f ( x ) = b }| = | A || B | .Perfect nonlinear functions have important applications in cryptogra-phy, sequences and coding theory. For example, as in [6], such functionscan be used to construct authentication codes.For the homomorphism φ m : ( Z /m Z ) ∗ → ( Z /m Z , +), defined inSection 4, we extend its definition to those integers a with gcd( a, m ) = 1by defining φ m ( a ) = 0. Then we get a function f m : ( Z /m Z , +) → ( Z /m Z , +) , x φ m ( x ) . For this function f m , we have the following proposition. Proposition 5.2. The function f m is perfect nonlinear if and only if m is a prime number.Proof. First, suppose that m is a prime number. By [6, Lemma 8] (or[5, Theorem 48]) and Proposition 4.3, it is easy to show that f m isperfect nonlinear.Now assume that m is a composite integer. Let p be a prime factor of m . Notice that f m ( kp ) = 0 for any k ≥ 1, and ( m +2) p ≤ m ( m +2) / References [1] T. Agoh, Fermat and Euler type quotients , C. R. Math. Rep. Acad. Sci.Canada, (1995), 159–164.[2] T. Agoh, On Giuga’s conjecture Manuscripta Math., (1995), 501–510.[3] T. Agoh, On Fermat and Wilson quotients , Expo. Math., (1996), 145–170.[4] T. Agoh, K. Dilcher and L. Skula, Fermat quotients for composite moduli , J.Number Theory, (1997), 29–50.[5] C. Carlet and C. Ding, Highly nonlinear mappings , J. Complexity (2004),205–244.[6] S. Chansona, C. Ding and A. Salomaab, Cartesian authentication codesfrom functions with optimal nonlinearity , Theoretical Computer Science, (2003), 1737–1752.[7] Z. Chen, Trace representation and linear complexity of binary sequences derivedfrom Fermat quotients , Sci. China Inf. Sci., (11) (2014), 1–10.[8] Z. Chen and X. Du, On the linear complexity of binary threshold sequencesderived from Fermat quotients , Des. Codes Cryptogr., (2013), 317–323. HE ARITHMETIC OF CARMICHAEL QUOTIENTS 17 [9] Z. Chen, A. Ostafe and A. Winterhof, Structure of pseudorandom numbersderived from Fermat quotients , in Arithmetic of Finite Fields , Lecture Notesin Computer Science , ed. by M.A. Hasan, T. Helleseth. vol. , pp. 73–85,Springer, Berlin, 2010.[10] Z. Chen and A. Winterhof, On the distribution of pseudorandom numbers andvectors derived from Euler-Fermat quotients , Int. J. Number Theory, (2012),631–641.[11] A. Granville, Some conjectures related to Fermat’s Last Theorem , in NumberTheory , ed. by R.A. Mollin, pp. 177–192, Walter de Gruyter, New York, 1990.[12] M. Lerch, Zur Theorie des Fermatschen Quotienten ( a p − − /p = q ( a ), Math.Ann., (1905), 471–490.[13] M. Lerch, Sur les th´eor`emes de Sylvester concernant le quotient de Fermat , C.R. Acad. Sci. Paris, (1906), 35–38.[14] F. Luca, M. Sha and I.E. Shparlinski, On two functions arising in the study ofthe Euler and Carmichael quotients , preprint, 2016.[15] P.L. Montgomery, New solutions of a p − ≡ p ), Math. Comp., (1993), 361–363.[16] A. Ostafe and I. Shparlinski, Pseudorandomness and dynamics of Fermat quo-tients , SIAM J. Discr. Math., (2011), 50–71.[17] P. Ribenboim, Thirteen lectures on Fermat’s Last Theorem , Springer, NewYork, 1979.[18] J. Sauerberg and L. Shu, Fermat quotients over function fields , Finite FieldsTh. App., (1997), 275–286.[19] L. Skula, Fermat and Wilson quotients for p -adic integers , Acta MathematicaUniversitatis Ostraviensis, (1998) , 167–181. School of Mathematics and Statistics, University of New SouthWales, Sydney, NSW 2052, Australia E-mail address ::