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The Gaussian zoo. (English) Zbl 1045.11071
Let \(A= \{a_{1}, \dots, a_{k}\}\) be a finite set of integers such that for every prime \(p\) the number of distinct entries in the mod \(p\) reduction of \(A\) is less than \(p.\) Then it is conjectured that there exist infinitely many integers \(n\) such that \(n + a_{1}, \dots, n + a_{k}\) are primes (“prime \(k\)-tuples conjecture”). Such sets \(A\) are called admissible. The authors determine all the maximal admissible connected sets of primes in Gaussian integers. The paper also contains an analogue of Mertens formula for Gaussian integers.

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11N80 Generalized primes and integers
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