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Totally indefinite Euclidean quaternion fields

Jean-Paul Cerri, Jérôme Chaubert, Pierre Lezowski (2014)

Acta Arithmetica

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.

Totally positive algebraic integers of small trace

Chistopher J. Smyth (1984)

Annales de l'institut Fourier

Let α be a totally positive algebraic integer, with the difference between its trace and its degree at most 6. We describe an algorithm for finding all such α , and display the resulting list of 1314 values of α which the algorithm produces.

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

Consider a recurrence sequence ( x k ) k of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a x k + 1 + a x k , where a , a , . . . , a n - 1 are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

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