On shifted primes with large prime factors and their products
نویسندگان
چکیده
منابع مشابه
Prime divisors of some shifted products
We study prime divisors of various sequences of positive integers A(n) + 1, n = 1,...,N, such that the ratios a(n) = A(n)/A(n − 1) have some number-theoretic or combinatorial meaning. In the case a(n) = n, we obviously have A(n) = n!, for which several new results about prime divisors of n! + 1 have recently been obtained.
متن کاملAdditive Functions on Shifted Primes
Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes. A real-valued function / defined on the positive integers is additive if it satisfies f(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determined by their values on the prime-powers. For additive arithmetic function /, let Q denote the frequency...
متن کاملLarge primes and Fermat factors
A systematic search for large primes has yielded the largest Fermat factors known.
متن کاملDivisors of Shifted Primes
Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend h...
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 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: Bulletin of the Belgian Mathematical Society - Simon Stevin
سال: 2015
ISSN: 1370-1444
DOI: 10.36045/bbms/1426856856