Common values of the arithmetic functions ϕ and σ
نویسندگان
چکیده
منابع مشابه
Common values of the arithmetic functions φ and σ Kevin
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 n solutions. The proofs rely on the recent work of the first two author...
متن کاملON COMMON VALUES OF φ(n) AND σ(m), II
For each positive-integer valued arithmetic function f , let Vf ⊂ N denote the image of f , and put Vf (x) := Vf ∩ [1, x] and Vf (x) := #Vf (x). Recently Ford, Luca, and Pomerance showed that Vφ ∩ Vσ is infinite, where φ denotes Euler’s totient function and σ is the usual sum-of-divisors function. Work of Ford shows that Vφ(x) ≍ Vσ(x) as x → ∞. Here we prove a result complementary to that of Fo...
متن کاملON COMMON VALUES OF φ(n) AND σ(m), I
We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x) numbers not exceeding x common to the ranges of φ and σ. Here φ is Euler’s totient function and σ is the sum-of-divisors function.
متن کاملCommon Values of the Arithmetic Functions
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 solutions, for some c > 0. The proofs rely on the recent work of the first two authors and K...
متن کاملφ and σ : from Euler to Erdős
This paper bij Florian Luca and Herman te Riele is an extended version of the 10th biennial Beeger Lecture, presented by Florian Luca on April 23, 2010 during the Nederlands Mathematisch Congres in Utrecht. Florian Luca (born 1969 in Romania) is full professor at the Instituto de Matemáticas, UNAM, Morelia, México. His research interests are abstract algebra, algebraic number theory, and Diopha...
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ژورنال
عنوان ژورنال: Bulletin of the London Mathematical Society
سال: 2010
ISSN: 0024-6093
DOI: 10.1112/blms/bdq014