Diophantine equations and Diophantine inequalities in algebraic number fields

1988 ◽  
pp. 83-94
Author(s):  
Yuan Wang
Keyword(s):  
Algebraic Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Diophantine Inequalities ◽  
Algebraic Number Fields

scholarly journals Diophantine equations and identities

1985 ◽  
Vol 8 (4) ◽  
pp. 755-777
Author(s):  
Malvina Baica
Keyword(s):  
Algebraic Number ◽  
Diophantine Equations ◽  
Number Fields ◽  
Combinatorial Identities ◽  
Common Method ◽  
Infinitely Many Solutions ◽  
Algebraic Number Fields ◽  
Pell’S Equation

The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are   i)  x2−my2=±1 ii)  x3+my3+m2z3−3mxyz=1iii)  Some fifth degree diopantine equationsInfinitely many solutions of each of these equations will be stated explicitly, using the results from the ACF discussed before.It is known that the solutions of Pell's equation are well exploited. We include it here because we shall use a common method to solve these three above mentioned equations and the method becomes very simple in Pell's equations case.Some new third and fifth degree combinatorial identities are derived from units in algebraic number fields.

scholarly journals A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

scholarly journals K-RATIONAL D-BRANE CRYSTALS

2012 ◽  
Vol 27 (22) ◽  
pp. 1250112
Author(s):  
ROLF SCHIMMRIGK
Keyword(s):  
String Theory ◽  
Algebraic Number ◽  
Special Class ◽  
Building Blocks ◽  
Number Fields ◽  
Central Point ◽  
Algebraic Number Fields ◽  
Special Values ◽  
Toroidal Compactifications ◽  
Base Points

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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