Akbal, YıldırımGüloğlu, Ahmet2023-02-212023-02-212022-07-051793-0421http://hdl.handle.net/11693/111566Let 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 .EnglishHardy–Littlewood circle methodr-free shifted primesGoldbach-type additive problemsArticle10.1142/S179304212350001X1793-7310