Exceptional units in a family of quartic number fields

Niklasch, Gerhard; Smart, Nigel P.

doi:10.1090/s0025-5718-98-00958-2

Cited by 13 publications

(11 citation statements)

References 16 publications

(24 reference statements)

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“…A unit u ∈ R is called exceptional if 1 − u ∈ R * . Since the solution of many Diophantine equations can be reduced to the solution of ax + by = 1 with x, y units in some ring, this amounts Communicated [10], Thue equations [20], Thue-Mahler equations [21], discriminant form equations [17] and so on (one can refer to [11,12]). …”

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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“…Example 8 (cf. [20]). Let t ≥ 4 and f t (x) = f (x) = x 4 + tx 3 + x 2 + tx − 1 = x(x 2 + 1)(x + t) − 1.…”

Section: Units In Fields With Signature (21)mentioning

confidence: 99%

Units in some families of algebraic number fields

Vulakh

2003

Trans. Amer. Math. Soc.

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“…Since then, they proved to be very beneficial when dealing with Diophantine equations of various types, e.g., for Thue equations [16] and ThueMahler equations [17] as demonstrated by Tzanakis and deWeger, discriminant form equations by Smart [13] and lots of others (for more references see [8]). The key idea is the fact that the solution of many Diophantine equations can be reduced to the solution of a finite number of unit equations of type ax + by = 1, where x and y are restricted to units in the ring of integers of some number field.…”

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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Exceptional units in a family of quartic number fields

Abstract: Abstract. We determine all exceptional units among the elements of certain groups of units in quartic number fields. These groups arise from a oneparameter family of polynomials with two real roots.

Cited by 13 publications

References 16 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

Units in some families of algebraic number fields

Sums of exceptional units in residue class rings

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