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p -adic Abelian Stark conjectures at s = 1

David Solomon (2002)

Annales de l’institut Fourier

A p -adic version of Stark’s Conjecture at s = 1 is attributed to J.-P. Serre and stated (faultily) in Tate’s book on the Conjecture. Building instead on our previous paper (and work of Rubin) on the complex abelian case, we give a new approach to such a conjecture for real ray-class extensions of totally real number fields. We study the coherence of our p -adic conjecture and then formulate some integral refinements, both alone and in combination with its complex analogue. A ‘Weak Combined Refined’ version...

Periodic Jacobi-Perron expansions associated with a unit

Brigitte Adam, Georges Rhin (2011)

Journal de Théorie des Nombres de Bordeaux

We prove that, for any unit ϵ in a real number field K of degree n + 1 , there exits only a finite number of n-tuples in  K n which have a purely periodic expansion by the Jacobi-Perron algorithm. This generalizes the case of continued fractions for n = 1 . For n = 2 we give an explicit algorithm to compute all these pairs.

Periods of sets of lengths: a quantitative result and an associated inverse problem

Wolfgang A. Schmid (2008)

Colloquium Mathematicae

The investigation of quantitative aspects of non-unique factorizations in the ring of integers of an algebraic number field gives rise to combinatorial problems in the class group of this number field. In this paper we investigate the combinatorial problems related to the function 𝓟(H,𝓓,M)(x), counting elements whose sets of lengths have period 𝓓, for extreme choices of 𝓓. If the class group meets certain conditions, we obtain the value of an exponent in the asymptotic formula of this function...

Polynomials of Pellian Type and Continued Fractions

Mollin, R. (2001)

Serdica Mathematical Journal

We investigate infinite families of integral quadratic polynomials {fk (X)} k∈N and show that, for a fixed k ∈ N and arbitrary X ∈ N, the period length of the simple continued fraction expansion of √fk (X) is constant. Furthermore, we show that the period lengths of √fk (X) go to infinity with k. For each member of the families involved, we show how to determine, in an easy fashion, the fundamental unit of the underlying quadratic field. We also demonstrate how the simple continued fraction ex-...

Prime factors of values of polynomials

J. Browkin, A. Schinzel (2011)

Colloquium Mathematicae

We prove that for every quadratic binomial f(x) = rx² + s ∈ ℤ[x] there are pairs ⟨a,b⟩ ∈ ℕ² such that a ≠ b, f(a) and f(b) have the same prime factors and min{a,b} is arbitrarily large. We prove the same result for every monic quadratic trinomial over ℤ.

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