Shifted primes without large prime factors

Divisors 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...

Squarefree Integers without Large Prime Factors in Short Intervals

Approximating the number of integers without large prime factors

Ψ(x, y) denotes the number of positive integers ≤ x and free of prime factors > y. Hildebrand and Tenenbaum gave a smooth approximation formula for Ψ(x, y) in the range (log x)1+ < y ≤ x, where is a fixed positive number ≤ 1/2. In this paper, by modifying their approximation formula, we provide a fast algorithm to approximate Ψ(x, y). The computational complexity of this algorithm is O( √ (log ...

Large primes and Fermat factors

A systematic search for large primes has yielded the largest Fermat factors known.

Additive 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...