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On additive bases II

Weidong Gao, Dongchun Han, Guoyou Qian, Yongke Qu, Hanbin Zhang (2015)

Acta Arithmetica

Let G be an additive finite abelian group, and let S be a sequence over G. We say that S is regular if for every proper subgroup H ⊆ G, S contains at most |H|-1 terms from H. Let ₀(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., every element of G can be expressed as the sum over a nonempty subsequence of S. The constant ₀(G) has been determined previously only for the elementary abelian groups. In this paper, we determine...

On generalized square-full numbers in an arithmetic progression

Angkana Sripayap, Pattira Ruengsinsub, Teerapat Srichan (2022)

Czechoslovak Mathematical Journal

Let a and b . Denote by R a , b the set of all integers n > 1 whose canonical prime representation n = p 1 α 1 p 2 α 2 p r α r has all exponents α i ( 1 i r ) ...

On the Davenport constant and group algebras

Daniel Smertnig (2010)

Colloquium Mathematicae

For a finite abelian group G and a splitting field K of G, let (G,K) denote the largest integer l ∈ ℕ for which there is a sequence S = g · . . . · g l over G such that ( X g - a ) · . . . · ( X g l - a l ) 0 K [ G ] for all a , . . . , a l K × . If (G) denotes the Davenport constant of G, then there is the straightforward inequality (G) - 1 ≤ (G,K). Equality holds for a variety of groups, and a conjecture of W. Gao et al. states that equality holds for all groups. We offer further groups for which equality holds, but we also give the first examples of groups G for which (G) -...

On the Diophantine equation x 2 - k x y + y 2 - 2 n = 0

Refik Keskin, Zafer Şiar, Olcay Karaatlı (2013)

Czechoslovak Mathematical Journal

In this study, we determine when the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 has an infinite number of positive integer solutions x and y for 0 n 10 . Moreover, we give all positive integer solutions of the same equation for 0 n 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 .

On the distribution of consecutive square-free primitive roots modulo p

Huaning Liu, Hui Dong (2015)

Czechoslovak Mathematical Journal

A positive integer n is called a square-free number if it is not divisible by a perfect square except 1 . Let p be an odd prime. For n with ( n , p ) = 1 , the smallest positive integer f such that n f 1 ( mod p ) is called the exponent of n modulo p . If the exponent of n modulo p is p - 1 , then n is called a primitive root mod p . Let A ( n ) be the characteristic function of the square-free primitive roots modulo p . In this paper we study the distribution n x A ( n ) A ( n + 1 ) , and give an asymptotic formula by using properties of character sums.

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