Variations on a theme of Mirsky

Let k and r be non-zero integers with r >= 2. An integer is called r-free if it is not divisible by the rth power of a prime. A result of Mirsky states that there are infinitely many primes p such that p + k is r-free. In this paper, we study an additive Goldbach-type problem and prove two uniform distribution results using these primes. We also study certain properties of primes p such that p + a1,....,p + al are simultaneously r-free, where a1,....,al are non-zero integers and l >= 1.