Approximating the number of integers without large prime factors

Upper bounds on the solutions to n = p+m^2

ardy and Littlewood conjectured that every large integer $n$ that is not a square is the sum of a prime and a square. They believed that the number $mathcal{R}(n)$ of such representations for $n = p+m^2$ is asymptotically given by begin{equation*} mathcal{R}(n) sim frac{sqrt{n}}{log n}prod_{p=3}^{infty}left(1-frac{1}{p-1}left(frac{n}{p}right)right), end{equation*} where $p$ is a prime, $m$ is a...

Approximating the number of integers free of large prime factors

Define Ψ(x, y) to be the number of positive integers n ≤ x such that n has no prime divisor larger than y. We present a simple algorithm that approximates Ψ(x, y) in O(y{ log log x log y + 1 log log y }) floating point operations. This algorithm is based directly on a theorem of Hildebrand and Tenenbaum. We also present data which indicate that this algorithm is more accurate in practice than o...

On Strings of Consecutive Integers with No Large Prime Factors

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ", we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding n, with the length of the strings tending to infinity with speed log log log log n. ...

On the Least Prime in Certain Arithmetic Progressions

We nd innnitely many pairs of coprime integers, a and q, such that the least prime a (mod q) is unusually large. In so doing we also consider the question of approximating rationals by other rationals with smaller and coprime denominators.

Squarefree Integers without Large Prime Factors in Short Intervals