On the Solutions of the Equation x m + y m - z m = 1 in a Finite Field

Wen, Kuei-Ann; Kiechle, Hubert

doi:10.2307/2161117

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Furthermore An Ith Basic Graph1

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Cited by 3 publications

(3 citation statements)

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“…Our aim is to show in which situations overlaps can occur in (F, Φ). The following two lemmas from [7] will be needed. The second case implies either p = 2, or that |Φ| is even.…”

Section: Furthermore An Ith Basic Graphmentioning

confidence: 99%

Combinatorial properties of ring generated circular planar nearrings

Wen

Kiechle

1996

Journal of Combinatorial Theory, Series A

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“…Our aim is to show in which situations overlaps can occur in (F, Φ). The following two lemmas from [7] will be needed. The second case implies either p = 2, or that |Φ| is even.…”

Section: Furthermore An Ith Basic Graphmentioning

confidence: 99%

Combinatorial properties of ring generated circular planar nearrings

Wen

Kiechle

1996

Journal of Combinatorial Theory, Series A

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“…√ q has been improved by using a number of methods; nevertheless, the exact value of N q is only known for few special cases [1,[7][8][9]12].…”

Section: Introductionmentioning

confidence: 99%

On the Number of Rational Points of Generalized Fermat Curves Over Finite Fields

Fanali

Giulietti

2012

Int. J. Number Theory

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The Stöhr-Voloch approach has been largely used to deal with the classical problem of estimating the number of rational points of a Fermat curve over a finite field. The same method actually applies to any curve admitting as an automorphism group the direct product of two cyclic groups C 1 and C 2 of the same size k, and such that the quotient curves with respect to both C 1 and C 2 are rational. In this paper such a curve is called a generalized Fermat curve. Our main achievement is that of extending some known results on Fermat curves to generalized Fermat curves. [5]. In [5], the non-singular models of plane curves with equation Remark 4.4. Comparison with the bounds inare investigated for k a divisor of q − 1. It is easily seen that Y k (a, b) is a generalized Fermat curve of order k for which case (B) holds. Theorem 3.6 in [5] gives some upper bounds on the number of F q -rational points of Y k (a, b). It should be noted that they are slightly better than inequality (1.2) here.

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“…[16−23] However, studies of producing atomic beams on the atom chip are scarce. Of course, freely expanded cold atoms can be used for the atom guiding, [19,21,23] but due to the freely expanding character, the wave packets of the guided atoms are spreading longitudinally, and the atom flux cannot be high. In order to solve this problem, static magnetic traps were used to accelerate and transfer atoms, but there need be extra controlling devices, such as macro magnetic coils, transport wires or extra laser beams.…”

Section: Introductionmentioning

confidence: 99%