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The iterated Carmichael λ function and the number of cycles of the power generator
, 2005
"... A common pseudorandom number generator is the power generator: x ↦ → x ℓ (mod n). Here, ℓ, n are fixed integers at least 2, and one constructs a pseudorandom sequence by starting at some residue mod n and iterating this ℓth power map. (Because it is the easiest to compute, one often takes ℓ = 2; thi ..."
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Cited by 6 (2 self)
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A common pseudorandom number generator is the power generator: x ↦ → x ℓ (mod n). Here, ℓ, n are fixed integers at least 2, and one constructs a pseudorandom sequence by starting at some residue mod n and iterating this ℓth power map. (Because it is the easiest to compute, one often takes ℓ = 2; this case is known as the BBS generator, for Blum,
The Lucas–Pratt primality tree
 Math. Comp
"... Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtai ..."
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Cited by 4 (0 self)
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Abstract. In 1876, E. Lucas showed that a quick proof of primality for a prime p could be attained through the prime factorization of p − 1 and a primitive root for p. V. Pratt’s proof that PRIMES is in NP, done via Lucas’s theorem, showed that a certificate of primality for a prime p could be obtained in O(log 2 p) modular multiplications with integers at most p. We show that for all constants C ∈ R, the number of modular multiplications necessary to obtain this certificate is greater than C log p for a set of primes p with relative asymptotic density 1. 1.
COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ
"... ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each ..."
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Cited by 4 (3 self)
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ABSTRACT. We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler’s totient function and σ is the sumofdivisors function. This proves a 50year old conjecture of Erdős. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c> 0. The proofs rely on the recent work of the first two authors and Konyagin on the distribution of primes p for which a given prime divides some iterate of φ at p, and on a result of HeathBrown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.
On some dynamical systems in finite fields and residue rings
 Discr. and Cont.Dynam.Syst.,Ser.A
"... We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to t ..."
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Cited by 2 (2 self)
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We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove them. We also show that some of them require further tuning. 1
Extremal orders of compositions of certain arithmetical functions
, 802
"... We study the exact extremal orders of compositions f(g(n)) of certain arithmetical functions, including the functions σ(n), φ(n), σ ∗ (n) and φ ∗ (n), representing the sum of divisors of n, Euler’s function and their unitary analogues, respectively. Our results complete, generalize and refine known ..."
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We study the exact extremal orders of compositions f(g(n)) of certain arithmetical functions, including the functions σ(n), φ(n), σ ∗ (n) and φ ∗ (n), representing the sum of divisors of n, Euler’s function and their unitary analogues, respectively. Our results complete, generalize and refine known results.
Primes in Classes of the Iterated Totient Function
"... As shown by Shapiro, the iterated totient function separates integers into classes having three sections. After summarizing some previous results about the iterated totient function, we prove five theorems about primes p in a class and the factorization of p − 1. An application of one theorem is the ..."
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As shown by Shapiro, the iterated totient function separates integers into classes having three sections. After summarizing some previous results about the iterated totient function, we prove five theorems about primes p in a class and the factorization of p − 1. An application of one theorem is the calculation of the smallest number in classes up to 1000. 1
PRIME CHAINS AND PRATT TREES
"... ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains wit ..."
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ABSTRACT. We study the distribution of prime chains, which are sequences p1,..., pk of primes for which pj+1 ≡ 1 (mod pj) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with pj+1 = 2pj +1 for each j. We give estimates for P (x), the number of chains with pk � x (k variable), and P (x; p), the number of chains with p1 = p and pk � px. The majority of the paper concerns the distribution of H(p), the length of the longest chain with pk = p, which is also the height of the Pratt tree for p. We show H(p) � c log log p and H(p) � (log p) 1−c′ for almost all p, with c, c ′ explicit positive constants. We can take, for any ε> 0, c = e − ε assuming the ElliottHalberstam conjecture. A stochastic model of the Pratt tree is introduced and analyzed. The model suggests that for most p � x, H(p) stays very close to e log log x. 1.
Smooth values of iterates of the Euler phifunction
 Canad. J. Math
"... Abstract. Let φ(n) be the Eulerphi function, define φ0(n) = n and φk+1(n) = φ(φk(n)) for all k ≥ 0. We will determine an asymptotic formula for the set of integers n less than x for which φk(n) is ysmooth, conditionally on a weak form of the ElliottHalberstam conjecture. Integers without large ..."
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Abstract. Let φ(n) be the Eulerphi function, define φ0(n) = n and φk+1(n) = φ(φk(n)) for all k ≥ 0. We will determine an asymptotic formula for the set of integers n less than x for which φk(n) is ysmooth, conditionally on a weak form of the ElliottHalberstam conjecture. Integers without large prime factors, usually called smooth numbers, play a central role in several topics of number theory. From multiplicative questions to analytic methods, they have various and wide applications, and understanding their behavior will have important consequences for number theoretic algorithms, which are an important tool in
Iterating the SumofDivisors Function
, 1995
"... this article denote positive integers, unless indicated otherwise, and oe denotes the sumofdivisors function. There is a great deal of literature concerning the iteration of the function oe(n) \Gamma n, much of it concerned with whether the iterated values eventually terminate at zero, cycle or be ..."
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this article denote positive integers, unless indicated otherwise, and oe denotes the sumofdivisors function. There is a great deal of literature concerning the iteration of the function oe(n) \Gamma n, much of it concerned with whether the iterated values eventually terminate at zero, cycle or become unbounded, depending on the value of n. See [Erdos et al. 1990; Guy 1994, p. 62] for details. Less work has been done on iterates of oe itself. We define oe
Compositions with the Euler and Carmichael Functions
"... Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1 ..."
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Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1