ON THE NUMBER OF DISTINCT PRIME FACTORS OF nj + ak
نویسنده
چکیده
Let ω(n) denote the number of distinct prime factors of n. Then for any given K ≥ 2, small ǫ > 0 and sufficiently large (only depending on K and ǫ) x, there exist at least x integers n ∈ [x, (1+K)x] such that ω(nj±ak) ≥ (log log log x) 1 3 −ǫ for all 2 ≤ a ≤ K, 1 ≤ j, k ≤ K and 0 ≤ h ≤ K log x.
منابع مشابه
On Covering Numbers
A positive integer n is called a covering number if there are some distinct divisors n1, . . . , nk of n greater than one and some integers a1, . . . , ak such that Z is the union of the residue classes a1(mod n1), . . . , ak(mod nk). A covering number is said to be primitive if none of its proper divisors is a covering number. In this paper we give some sufficient conditions for n to be a (pri...
متن کاملOn some Frobenius groups with the same prime graph as the almost simple group ${ {bf PGL(2,49)}}$
The prime graph of a finite group $G$ is denoted by $Gamma(G)$ whose vertex set is $pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $Gamma(H)=Gamma(G)$, in while $Hnotcong G$. In this paper, we consider finite groups with the same prime gr...
متن کاملThe distribution of totatives
It is clear that Ak and Bk are subsets of Ck, and in fact Lehmer [3] proved that Ak ⊂ Bk ⊂ Ck. It is not difficult to show that Cp ⊂ Ap for a prime p, so that Ap = Bp = Cp. P. J. McCarthy [5] proved that Ak = Bk when k is not squarefree, and he asked if the result could be extended to all composite numbers k. This was done by P. Erdős [2], who proved that the set Bk \ Ak is infinite for every c...
متن کاملResidue races by Andrew Granville , Daniel Shiu and Peter Shiu
Given a prime p and distinct non-zero integers a1, a2, . . . , ak (mod p), we investigate the number N = N(a1, a2, . . . , ak; p) of residues n (mod p) for which (na1)p < (na2)p < . . . < (nak)p, where (b)p is the least non-negative residue of b (mod p). We give complete solutions to the problem when k = 2, 3, 4, and establish some general results corresponding to k ≥ 5. 2000 Mathematics Subjec...
متن کاملA note on the order graph of a group
The order graph of a group $G$, denoted by $Gamma^*(G)$, is a graph whose vertices are subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $|H|big{|}|K|$ or $|K|big{|}|H|$. In this paper, we study the connectivity and diameter of this graph. Also we give a relation between the order graph and prime graph of a group.
متن کامل