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Arithmetic progressions in a unique factorization domain. (English) Zbl 1256.11008

The generalized Pillai theorem states that in any set of at most 16 consecutive integers in arithmetic progression, there exists an integer that is relatively prime to all the rest. Here the authors are interested in the analogue of this result to arbitrary integral domains where the notion of greatest common divisor makes sense.
The main result is an analogue of the generalized Pillai theorem for the so-called \(\sigma\)-atomic GCD domains of characteristic zero and in particular, for arbitrary unique factorization domains of characteristic zero. E.g., the generalized Pillai theorem holds for the Gaussian integers with 16 replaced by 6.

MSC:

11B25 Arithmetic progressions
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
13A05 Divisibility and factorizations in commutative rings

Citations:

Zbl 1248.11072
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