On the difference between the number of prime divisors at consecutive integers

1999 ◽  
Vol 66 (4) ◽  
pp. 474-488
Author(s):  
N. M. Timofeev
Keyword(s):  
Prime Divisors ◽  
Consecutive Integers ◽  
The Difference ◽  
Number Of Prime Divisors

scholarly journals Number of prime divisors in a product of consecutive integers

2004 ◽  
Vol 113 (4) ◽  
pp. 327-341 ◽  
Author(s):  
Shanta Laishram ◽  
T. N. Shorey
Keyword(s):  
Prime Divisors ◽  
Consecutive Integers ◽  
Number Of Prime Divisors

scholarly journals On the number of prime divisors of the order of elliptic curves modulo p

2005 ◽  
Vol 117 (4) ◽  
pp. 341-352 ◽  
Author(s):  
Jörn Steuding ◽  
Annegret Weng
Keyword(s):  
Elliptic Curves ◽  
Prime Divisors ◽  
Modulo P ◽  
Number Of Prime Divisors

scholarly journals The distribution of the number of prime divisors of sums a + b

1988 ◽  
Vol 29 (1) ◽  
pp. 94-99 ◽  
Author(s):  
P.D.T.A Elliott ◽  
A Sárközy
Keyword(s):  
Prime Divisors ◽  
Number Of Prime Divisors

On the Normal Growth of Prime Factors of Integers

1992 ◽  
Vol 44 (6) ◽  
pp. 1121-1154 ◽  
Author(s):  
J. M. De Koninck ◽  
I. Kátai ◽  
A. Mercier
Keyword(s):  
Prime Factor ◽  
Continuous Distribution ◽  
Normal Growth ◽  
Distribution Functions ◽  
Prime Divisors ◽  
Simple Argument ◽  
Prime Factors ◽  
Almost All ◽  
Number Of Prime Divisors ◽  
Class Of Functions

AbstractLet h: [0,1] → R be such that and define .In 1966, Erdős [8] proved that holds for almost all n, which by using a simple argument implies that in the case h(u) = u, for almost all n, He further obtained that, for every z > 0 and almost all n, and that where ϕ, ψ, are continuous distribution functions. Several other results concerning the normal growth of prime factors of integers were obtained by Galambos [10], [11] and by De Koninck and Galambos [6].Let χ = ﹛xm : w ∈ N﹜ be a sequence of real numbers such that limm→∞ xm = +∞. For each x ∈ χ let be a set of primes p ≤x. Denote by p(n) the smallest prime factor of n. In this paper, we investigate the number of prime divisors p of n, belonging to for which Th(n,p) > z. Given Δ < 1, we study the behaviour of the function We also investigate the two functions , where, in each case, h belongs to a large class of functions.

A NOTE ON THE ERDŐS–GRAHAM THEOREM

2018 ◽  
Vol 97 (3) ◽  
pp. 363-366
Author(s):  
WENHUI WANG ◽  
MIN TANG
Keyword(s):  
Large Integer ◽  
Consecutive Integers ◽  
The Difference ◽  
Asymptotic Basis

Let ${\mathcal{A}}=\{a_{1}<a_{2}<\cdots \,\}$ be a set of nonnegative integers. Put $D({\mathcal{A}})=\gcd \{a_{k+1}-a_{k}:k=1,2,\ldots \}$. The set ${\mathcal{A}}$ is an asymptotic basis if there exists $h$ such that every sufficiently large integer is a sum of at most $h$ (not necessarily distinct) elements of ${\mathcal{A}}$. We prove that if the difference of consecutive integers of ${\mathcal{A}}$ is bounded, then ${\mathcal{A}}$ is an asymptotic basis if and only if there exists an integer $a\in {\mathcal{A}}$ such that $(a,D({\mathcal{A}}))=1$.

scholarly journals On the maximal length of two sequences of consecutive integers with the same prime divisors

1989 ◽  
Vol 54 (3-4) ◽  
pp. 225-236 ◽  
Author(s):  
R. Balasubramanian ◽  
T. N. Shorey ◽  
M. Waldschmidt
Keyword(s):  
Maximal Length ◽  
Prime Divisors ◽  
Consecutive Integers
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