# zbMATH — the first resource for mathematics

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 $$k^2+k\log_2n-n/4\leq0$$.

##### MSC:
 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
Full Text: