Year: 2010
Type: article
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 fifty-year-old conjecture of Erdős. Moreover, we show that, for some c > 0, there are infinitely many integers n such that ϕ(a) = n and σ(b) = n, each having more than nc solutions. 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.
DOI: 10.1112/blms/bdq014
Authors: Kevin Ford, Florian Luca, Carl Pomerance
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1863786576