# Divisors of Shifted Primes

@article{Koukoulopoulos2010DivisorsOS, title={Divisors of Shifted Primes}, author={Dimitris Koukoulopoulos}, journal={International Mathematics Research Notices}, year={2010}, volume={2010}, pages={4585-4627} }

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 heavily on our knowledge about primes in arithmetic progressions. As an application of these bounds, we determine…

## 19 Citations

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Abstract We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$ . Our estimate is uniform in $x,y,w$ . We apply…

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- MathematicsCanadian Journal of Mathematics
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Abstract We prove a new bound on collinear triples in subgroups of prime finite fields and use it to give some new bounds on exponential sums with trinomials.

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