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30
Irreducible radical extensions and Eulerfunction chains, Combinatorial number theory
 Proc. Integers Conf. 2005), 351361, de Gruyter
, 2007
"... For Ron Graham on his 70th birthday We discuss the smallest algebraic number eld which contains the nth roots of unity and which may be reached from the rational eld Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this eld over Q is '(m), where m is the smallest m ..."
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For Ron Graham on his 70th birthday We discuss the smallest algebraic number eld which contains the nth roots of unity and which may be reached from the rational eld Q by a sequence of irreducible, radical, Galois extensions. The degree D(n) of this eld over Q is '(m), where m is the smallest multiple of n divisible by each prime factor of '(m). The prime factors of m=n are precisely the primes not dividing n but which do divide some number in the \Euler chain " '(n); '('(n)); : : :. For each xed k, we show that D(n)> nk on a set of asymptotic density 1.
The iterated Carmichael λ function and the number of cycles of the power generator
 Acta Arith
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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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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 the distribution of sociable numbers
 J. Number Theory
, 2009
"... Abstract. For a positive integer n, dene s(n) as the sum of the proper divisors of n. If s(n)> 0, dene s2(n) = s(s(n)), and so on for higher iterates. Sociable numbers are those n with sk(n) = n for some k, the least such k being the order of n. Such numbers have been of interest since antiquit ..."
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Abstract. For a positive integer n, dene s(n) as the sum of the proper divisors of n. If s(n)> 0, dene s2(n) = s(s(n)), and so on for higher iterates. Sociable numbers are those n with sk(n) = n for some k, the least such k being the order of n. Such numbers have been of interest since antiquity, when order1 sociables (perfect numbers) and order2 sociables (amicable numbers) were studied. In this paper we make progress towards the conjecture that the sociable numbers have asymptotic density 0. We show that the number of sociable numbers in [1; x], whose cycle contains at most k numbers greater than x, is o(x) for each xed k. In particular, the number of sociable numbers whose cycle is contained entirely in [1; x] is o(x), as is the number of sociable numbers in [1; x] with order at most k. We also prove that but for a set of sociable numbers of asymptotic density 0, all sociable numbers are contained within the set of odd abundant numbers, which has asymptotic density about 1=500.
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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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
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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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.
Some normal numbers generated by arithmetic functions, submitted; preprint available online as arXiv:1309.7386
"... Abstract. Let g ≥ 2. A real number is said to be gnormal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕ denote Euler’s totient function, let σ be the sumofdivisors function, and let λ be Carmichael’s lambdafunction. We show that if f ..."
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Abstract. Let g ≥ 2. A real number is said to be gnormal if its base g expansion contains every finite sequence of digits with the expected limiting frequency. Let ϕ denote Euler’s totient function, let σ be the sumofdivisors function, and let λ be Carmichael’s lambdafunction. We show that if f is any function formed by composing ϕ, σ, or λ, then the number 0.f(1)f(2)f(3)... obtained by concatenating the base g digits of successive fvalues is gnormal. We also prove the same result if the inputs 1, 2, 3,... are replaced with the primes 2, 3, 5,.... The proof is an adaptation of a method introduced by Copeland and Erdős in 1946 to prove the 10normality of 0.235711131719.... 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.
On the Distribution of Perfect Totients
, 2006
"... In this paper, we study the sum of iterates of the Euler function. ..."
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In this paper, we study the sum of iterates of the Euler function.
SIEVING VERY THIN SETS OF PRIMES, AND PRATT TREES WITH MISSING PRIMES
"... ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. O ..."
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ABSTRACT. Suppose P is a set of primes, such that for every p ∈ P, every prime factor of p − 1 is also in P. We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x 1−c) for some c> 0, where c depends only on the smallest prime not in P. Our proof makes use of results connected with Artin’s primitive root conjecture. 1