Prime divisors of Lucas sequences
Pieter Moree, Peter Stevenhagen (1997)
Acta Arithmetica
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Pieter Moree, Peter Stevenhagen (1997)
Acta Arithmetica
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David S. Dummit, Jonathan W. Sands, Brett Tangedal (2003)
Journal de théorie des nombres de Bordeaux
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Stark’s conjectures connect special units in number fields with special values of -functions attached to these fields. We consider the fundamental equality of Stark’s refined conjecture for the case of an abelian Galois group, and prove it when this group has exponent . For biquadratic extensions and most others, we prove more, establishing the conjecture in full.
J. E. Carroll, H. Kisilevsky (1976)
Compositio Mathematica
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Pieter Moree (1996)
Journal de théorie des nombres de Bordeaux
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The density of primes dividing at least one term of the Lucas sequence , defined by and for , with an arbitrary integer, is determined.
Ken Yamamura (1997)
Journal de théorie des nombres de Bordeaux
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We determine the structures of the Galois groups Gal of the maximal unramified extensions of imaginary quadratic number fields of conductors under the Generalized Riemann Hypothesis). For all such , is , the Hilbert class field of , the second Hilbert class field of , or the third Hilbert class field of . The use of Odlyzko’s discriminant bounds and information on the structure of class groups obtained by using the action of Galois groups on class groups is essential. We...
Ken Yamamura (2001)
Journal de théorie des nombres de Bordeaux
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In the previous paper [15], we determined the structure of the Galois groups of the maximal unramified extensions of imaginary quadratic number fields of conductors under the Generalized Riemann Hypothesis (GRH) except for 23 fields (these are of conductors ) and give a table of . We update the table (under GRH). For 19 exceptional fields of them, we determine . In particular, for , we obtain , the fourth Hilbert class field of . This is the first example of a number...
A. Vazzana (1997)
Acta Arithmetica
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1. Introduction. For quadratic fields whose discriminant has few prime divisors, there are explicit formulas for the 4-rank of . For quadratic fields whose discriminant has arbitrarily many prime divisors, the formulas are less explicit. In this paper we will study fields of the form , where the primes are all congruent to 1 mod 8. We will prove a theorem conjectured by Conner and Hurrelbrink which examines under what conditions the 4-rank of is zero for such fields. In the course...
P. Elliott (1968)
Acta Arithmetica
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Franz Halter-Koch (2003)
Journal de théorie des nombres de Bordeaux
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Let be a set of binary quadratic forms of the same discriminant, a set of arithmetical progressions and a positive integer. We investigate the representability of prime powers lying in some progression from by some form from .
P. Elliott (1967)
Acta Arithmetica
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Pierre Kaplan, Kenneth Williams, Yoshihiko Yamamoto (1984)
Acta Arithmetica
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Franz Lemmermeyer (1997)
Journal de théorie des nombres de Bordeaux
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Classical results of Rédei, Reichardt and Scholz show that unramified cyclic quartic extensions of quadratic number fields correspond to certain factorizations of its discriminant disc . In this paper we extend their results to unramified quaternion extensions of which are normal over , and show how to construct them explicitly.