Results 1 
4 of
4
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 7 (6 self)
 Add to MetaCart
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.
RESULTS CONCERNING AN EQUATION OF GOORMAGHTIGH AND RELATED TOPICS
"... Y. Bugeaud and T. N. Shorey [1] studied the equation x u−1 x−1 = yw−1 y−1 in integers y> x> 1, u> 1, and w> 1. They showed that, if you fix the values of x and y, there are at most two solutions to this equation. The focus of the current paper is on exploring properties of the exponents ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Y. Bugeaud and T. N. Shorey [1] studied the equation x u−1 x−1 = yw−1 y−1 in integers y> x> 1, u> 1, and w> 1. They showed that, if you fix the values of x and y, there are at most two solutions to this equation. The focus of the current paper is on exploring properties of the exponents if two solutions exist. The equation σ(x)/x = σ(pm1qn1)/(pm1qn1) and relationships between (xu − 1)/(x − 1) = (yw − 1)/(y − 1) and σ(pm1qn1)/(pm1qn1) = σ(pm2qn2)/(pm2qn2) are also discussed, where p and q represent prime numbers, m1, n1, m2, and n2 are positive integers, and pm1qn1 = pm2qn2.
Solving the Odd Perfect Number Problem: Some Old and New Approaches
, 2008
"... ar ..."
(Show Context)