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Residue classes free of values of Euler’s function
 In: Gy}ory K (ed) Proc Number Theory in Progress, pp 805–812. Berlin: W de Gruyter
, 1999
"... Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains ..."
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Dedicated to Andrzej Schinzel on his sixtieth birthday By a totient we mean a value taken by Euler’s function φ(n). Dence and Pomerance [DP] have established Theorem A. If a residue class contains at least one multiple of 4, then it contains
Smooth Orders and Cryptographic Applications
 Lecture Notes in Comptuer Science
, 2002
"... We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker re ..."
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Cited by 5 (1 self)
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We obtain rigorous upper bounds on the number of primes x for which p1 is smooth or has a large smooth factor. Conjecturally these bounds are nearly tight. As a corollary, we show that for almost all primes p the multiplicative order of 2 modulo p is not smooth, and we prove a similar but weaker result for almost all odd numbers n. We also discuss some cryptographic applications.
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.
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
Article 13.8.8 On the Ratio of the Sum of Divisors and Euler’s Totient Function I
"... We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) ..."
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We prove that the only solutions to the equation σ(n) = 2 · φ(n) with at most three distinct prime factors are 3, 35 and 1045. Moreover, there exist at most a finite number of solutions to σ(n) = 2·φ(n) with Ω(n) ≤ k, and there are at most 22k +k −k squarefree solutions to φ(n) ∣ ∣σ(n) if ω(n) = k. Lastly the number of solutions to φ(n) ∣ ∣σ(n) as x → ∞ is O ( xexp ( −1 √)) 2 logx. 1