منابع مشابه
Sieving the Positive Integers by Large Primes
Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if y-t cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous solution of the differential delay equation up&(u) = -6p,(u 1 ), p&(u) = 1, 0 < I( < 1. This generalizes wo...
متن کاملSieving the Positive Integers by Small Primes
Let Q be a set of primes tha t has relative density 6 among the primes, and let 4 (x,y, Q ) be the number of positive integers 5 x tha t have no prime factor 5 y from the set Q. Standard sieve methods do not seem to give an asymptotic formula for 4 (x,y , Q ) in the case tha t 5 6 < 1. We use a method of Hildebrand to prove that as x -+ w, where u = !S2 and f ( u ) is defined by log Y This may ...
متن کاملSieving by Large Integers and Covering Systems of Congruences
An old question of Erdős asks if there exists, for each number N , a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if ∑ n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, f...
متن کاملIntegers, Prime Factorization, and More on Primes
The integer q is called the quotient and r is the remainder. Proof. Consider the rational number b a . Since R = ⋃ k∈Z[k, k + 1) (disjoint), there exists a unique integer q such that b a ∈ [q, q + 1), i.e., q ≤ b a < q + 1. Multiplying through by the positive integer a, we obtain qa ≤ b < (q + 1)a. Let r = b− qa. Then we have b = qa + r and 0 ≤ r < a, as required. Proposition 3. Let a, b, d ∈ Z...
متن کاملSieving Very Thin Sets of Primes, and Pratt Trees with Missing Primes
Suppose P is a set of primes, such that for every p ∈ P , every prime factor of p − 1 is also in P . We apply a new sieve method to show that either P contains all of the primes or the counting function of P is O(x) for some c > 0, where c depends only on the smallest prime not in P . Our proof makes use of results connected with Artin’s primitive root conjecture.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Mathematics Research
سال: 2019
ISSN: 1916-9809,1916-9795
DOI: 10.5539/jmr.v11n1p17