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A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

Weidong Gao, Jiangtao Peng, Qinghai Zhong (2013)

Acta Arithmetica

Let K be an algebraic number field with non-trivial class group G and K be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let F k ( x ) denote the number of non-zero principal ideals a K with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that F k ( x ) behaves for x → ∞ asymptotically like x ( l o g x ) 1 - 1 / | G | ( l o g l o g x ) k ( G ) . We prove, among other results, that ( C n C n ) = n + n for all integers n₁,n₂ with 1 < n₁|n₂.

A recursive definition of p -ary addition without carry

François Laubie (1999)

Journal de théorie des nombres de Bordeaux

Let p be a prime number. In this paper we prove that the addition in p -ary without carry admits a recursive definition like in the already known cases p = 2 and p = 3 .

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

For k = 1,2,... let H k denote the harmonic number j = 1 k 1 / j . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have k = 1 p - 1 ( H k ) / ( k 2 k ) 7 / 24 p B p - 3 ( m o d p ² ) , k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) - 3 / 8 B p - 3 ( m o d p ) , and k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and H k , m : = j = 1 k 1 / ( j m ) .

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