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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 ₁ ( C n ₁ ⊕ C n ₂ ) = n ₁ + n ₂ for all integers n₁,n₂ with 1 < n₁|n₂.

A quantitative primitive divisor result for points on elliptic curves

Patrick Ingram (2009)

Journal de Théorie des Nombres de Bordeaux

Let E / K be an elliptic curve defined over a number field, and let P ∈ E ( K ) be a point of infinite order. It is natural to ask how many integers n ≥ 1 fail to occur as the order of P modulo a prime of K . For K = ℚ , E a quadratic twist of y 2 = x 3 - x , and P ∈ E ( ℚ ) as above, we show that there is at most one such n ≥ 3 .

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 .

A search for Tribonacci-Wieferich primes

Jiří Klaška (2008)

Acta Mathematica Universitatis Ostraviensis

Such problems as the search for Wieferich primes or Wall-Sun-Sun primes are intensively studied and often discused at present. This paper is devoted to a similar problem related to the Tribonacci numbers.

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