COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ

by Kevin Ford , Florian Luca , Carl Pomerance

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Datum	Value	Source
TITLE	COMMON VALUES OF THE ARITHMETIC FUNCTIONS φ AND σ	SVM HeaderParse 0.2
AUTHOR NAME	Kevin Ford	SVM HeaderParse 0.2
AUTHOR NAME	Florian Luca	SVM HeaderParse 0.2
AUTHOR NAME	Carl Pomerance	SVM HeaderParse 0.2
ABSTRACT	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 sum-of-divisors function. This proves a 50-year 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 Heath-Brown connecting the possible existence of Siegel zeros with the distribution of twin primes. 1.	SVM HeaderParse 0.2
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