Results 1  10
of
16
The number of solutions of Φ(x) = m
"... An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an uncondit ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
An old conjecture of Sierpiński asserts that for every integer k � 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler’s totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpiński’s conjecture. The proof uses many results from sieve theory, in particular the famous theorem of Chen.
An uncertainty principle for arithmetic sequences
, 2004
"... Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Analytic number theorists usually seek to show that sequences which appear naturally in arithmetic are “welldistributed ” in some appropriate sense. In various discrepancy problems, combinatorics researchers have analyzed limitations to equidistribution, as have Fourier analysts when working with the “uncertainty principle”. In this article we find that these ideas have a natural setting in the analysis of distributions of sequences in analytic number theory, formulating a general principle, and giving several examples.
On Goldbach’s conjecture in arithmetic progressions
 Stud. Sci. Math. Hungar
"... ABSTRACT. It is proved that for a given integer N and for all but (log N)B prime numbers k ≤ N5/48−ε the following is true: For any positive integers bi, i ∈ {1, 2, 3}, (bi, k) = 1 that satisfy N ≡ b1 + b2 + b3 (mod k), N can be written as N = p1+p2+p3, where the pi, i ∈ {1, 2, 3} are prime number ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. It is proved that for a given integer N and for all but (log N)B prime numbers k ≤ N5/48−ε the following is true: For any positive integers bi, i ∈ {1, 2, 3}, (bi, k) = 1 that satisfy N ≡ b1 + b2 + b3 (mod k), N can be written as N = p1+p2+p3, where the pi, i ∈ {1, 2, 3} are prime numbers that satisfy pi ≡ bi (mod k). 1. Introduction. Vinogradov [17] has proved that every sufficiently large odd positive integer can be written as the sum of three primes. This theorem has been generalized in many ways. In 1953, Ayoub [1] proved the following result: If k is a fixed positive integer, bi, i = 1, 2, 3, are integers with (bi, k) = 1 and J(N; k, b1, b2, b3) is the number of
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 4 (3 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.
An improvement on a theorem of the GoldbachWaring type
 Rocky Mountain Journal of Mathematics
"... ABSTRACT. Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but x19193/19200+ε positive even integers N smaller than x can be represented as N = p21 + p 3 2 + p ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. Let pi, 2 ≤ i ≤ 5 be prime numbers. It is proved that all but x19193/19200+ε positive even integers N smaller than x can be represented as N = p21 + p 3 2 + p
SUMS OF PRIMES AND SQUARES OF PRIMES IN SHORT INTERVALS
, 801
"... Abstract. Let H2 denote the set of even integers n ̸ ≡ 1 (mod 3). We prove that when H ≥ X 0.33, almost all integers n ∈ H2 ∩ (X, X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes. 1. ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. Let H2 denote the set of even integers n ̸ ≡ 1 (mod 3). We prove that when H ≥ X 0.33, almost all integers n ∈ H2 ∩ (X, X + H] can be represented as the sum of a prime and the square of a prime. We also prove a similar result for sums of three squares of primes. 1.
ADDITIVE PROBLEMS WITH PRIME VARIABLES THE CIRCLE METHOD OF HARDY, RAMANUJAN AND LITTLEWOOD
, 2012
"... In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sket ..."
Abstract
 Add to MetaCart
(Show Context)
In these lectures we give an overview of the circle method introduced by Hardy and Ramanujan at the beginning of the twentieth century, and developed by Hardy, Littlewood and Vinogradov, among others. We also try and explain the main difficulties in proving Goldbach’s conjecture and we give a sketch of the proof of Vinogradov’s threeprime Theorem.