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On a combinatorial problem connected with factorizations. (English) Zbl 0867.11075
Let \(R\) be the ring of integers of an algebraic number field and denote by \(a_k\) the maximal number of non-principal prime ideals which can divide a squarefree element of \(R\) having at most \(k\) distinct factorizations into irreducibles. It is known [W. Narkiewicz, Colloq. Math. 42, 319-330 (1979; Zbl 0514.12004)] that \(a_k=a_k(H)\) depends only on \(k\) and the class group \(H\) of \(R\) and one can give a purely combinatorial definition of it. This constant appears in the asymptotic formula for the number of principal ideals of \(R\) with norms \(\leq x\), whose generators have at most \(k\) distinct factorizations into irreducibles. The explicit determination of \(a_k\) presents difficulties, and the author succeeds in showing that for several classes of groups \(H=\bigoplus _{j=1}^rC_{n_j}\) (with \(n_1|n_2|\cdots|n_r)\) the constant \(a_1(H)\) equals \(\sum n_j\). He determines also the value of \(a_k\) for \(k\geq2\) and cyclic groups of \(n\) elements, provided \(k^2+k\log_2n-n/4\leq0\).

11R27 Units and factorization
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups
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