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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.
Article 11.5.5 On the Solutions of σ(n) = σ(n + k)
"... Given a fixed even integer k, we show that Schinzel’s hypothesis H implies that σ(n) = σ(n + k) infinitely often. We also discuss the case of odd k and the more general equation σα(n) = σα(n + k). 1 ..."
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Given a fixed even integer k, we show that Schinzel’s hypothesis H implies that σ(n) = σ(n + k) infinitely often. We also discuss the case of odd k and the more general equation σα(n) = σα(n + k). 1