On cliques of exceptional units and Lenstra's construction of Euclidean fields

Leutbecher, Armin; Niklasch, Gerhard

doi:10.1007/bfb0086551

Cited by 15 publications

(10 citation statements)

References 17 publications

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“…After that, many new Euclidean number fields were found (see [6,9]). Moreover, exceptional units also have connections with cyclic resultants [18,19] and Lehmer's conjecture related to Mahler's measure [15,16].…”

Section: Introductionmentioning

confidence: 99%

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

Yang

Zhao

2016

Monatsh Math

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Let R be a commutative ring with 1 ∈ R and R * be the multiplicative group of its units. In 1969, Nagell introduced the concept of an exceptional unit, namely a unit u such that 1 − u is also a unit. Let Z n be the ring of residue classes modulo n. In this paper, given an integer k ≥ 2, we obtain an exact formula for the number of ways to represent each element of Z n as the sum of k exceptional units. This generalizes a recent result of J. W. Sander for the case k = 2.

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Section: Introductionmentioning

confidence: 99%

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

Yang

Zhao

2016

Monatsh Math

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“…By further development of this method, quite a few formerly unknown Euclidean number fields could be found by Leutbecher and Niklasch [5] and Houriet [3]. Exunits were also studied for their own sake, e.g., the calculation of the number of exunits in a number field of given degree and unit rank [7].…”

Section: Introductionmentioning

confidence: 99%

Sums of exceptional units in residue class rings

Sander

2016

Journal of Number Theory

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Given a commutative ring R with 1 ∈ R and the multiplicative group R * of units, an element u ∈ R * is called an exceptional unit if 1 − u ∈ R * , i.e., if there is a u ∈ R * such that u + u = 1. We study the case R = Z n := Z/nZ of residue classes mod n and determine the number of representations of an arbitrary element c ∈ Z n as the sum of two exceptional units. As a consequence, we obtain the sumset Z * * n +Z * * n for all positive integers n, with Z * * n denoting the set of exceptional units of Z n .

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“…In Győry [18], [21], [22], [23], [26] the structure of these graphs was described from the point of view of connectedness. For related results and applications, we refer to Győry [16], [19], [20], [25], Evertse, Győry, Stewart and Tijdeman [9], Leutbecher [33], Leutbecher and Niklash [34], Győry, Hajdu and Tijdeman [27], Ruzsa [36] and the references there.…”

Section: Introductionmentioning

confidence: 99%

Representation of finite graphs as difference graphs of S-units, I

Győry

Hajdu

Tijdeman

2014

Journal of Combinatorial Theory, Series A

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Abstract. Let S be a finite non-empty set of primes, Z S the ring of those rationals whose denominators are not divisible by primes outside S, and Z * S the multiplicative group of invertible elements (S-units) in Z S . For a non-empty subset A of Z S , denote by G S (A) the graph with vertex set A and with an edge between a and b if and only if a − b ∈ Z * S . This type of graphs has been studied by many people.In the present paper we deal with the representability of finite (simple) graphs G as G S (A). If A = uA + a for some u ∈ Z * S and a ∈ Z S , then A and A are called S-equivalent, since G S (A) and G S (A ) are isomorphic. We say that a finite graph G is representable / infinitely representable with S if G is isomorphic to G S (A) for some A / for infinitely many non-S-equivalent A.We prove among other things that for any finite graph G there exist infinitely many finite sets S of primes such that G can be represented with S. We deal with the infinite representability of finite graphs, in particular cycles and complete bipartite graphs. Further, we consider the triangles in G for a deeper analysis. Finally, we prove that G is representable with every S if and only if G is cubical.Besides combinatorial and numbertheoretical arguments, some deep Diophantine results concerning S-unit equations are used in our proofs.In Part II, we shall investigate these and similar problems over more general domains.2010 Mathematics Subject Classification. 05C25, 05C62, 11D61.

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On cliques of exceptional units and Lenstra's construction of Euclidean fields

Cited by 15 publications

References 17 publications

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

On the sumsets of exceptional units in $$\mathbb {Z}_n$$ Z n

Sums of exceptional units in residue class rings

Representation of finite graphs as difference graphs of S-units, I

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