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On values taken by the largest prime factor of shifted primes
 Journal of the Australian Mathematical Society
"... Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c( ..."
Abstract

Cited by 5 (2 self)
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Let P denote the set of prime numbers, and let P(n) denote the largest prime factor of an integer n> 1. We show that, for every real number 32/17 < η < (4 + 3 √ 2)/4, there exists a constant c(η)> 1 such that for every integer a � = 0, the set � p ∈ P: p = P(q − a) for some prime q with p η < q < c(η) p η � has relative asymptotic density one in the set of all prime numbers. Moreover, in the range 2 ≤ η < (4+3 √ 2)/4, one can take c(η) = 1+ε for any fixed ε> 0. In particular, our results imply that for every real number 0.486 ≤ ϑ ≤ 0.531, the relation P(q − a) ≍ q ϑ holds for infinitely many primes q. We use this result to derive a lower bound on the number of distinct prime divisors of the value of the Carmichael function taken on a product of shifted primes. Finally, we study iterates of the map q ↦ → P(q − a) for a> 0, and show that for infinitely many primes q, this map can be iterated at least (log log q) 1+o(1) times before it terminates. 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 ..."
Abstract

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.