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Abstract

The investigation of the counting function of the set of integral elements, in an algebraic number field, with factorizations of at most $k$ different lengths gives rise to a combinatorial constant depending only on the class group of the number field and the integer $k$. In this paper the value of these constants, in case the class group is an elementary $p$-group, is estimated, and determined under additional conditions. In particular, it is proved that for elementary $2$-groups these constants are equivalent to constants that are investigated in extremal graph theory.