Attila Pethő

Thomas' Family of Thue Equations Over Imaginary Quadratic Fields

We consider the family of relative Thue equations x3? (t? 1)x2y? (t+ 2)xy2?y3= ?,where the parameter t, the root of unity? and the solutions x and y are integers in the same imaginary quadratic number field.We prove that there are only trivial solutions (with x, y? 1), if t is large enough or if the discriminant of the quadratic number field is large enough or ifRet=?1/2 (there are a few more solutions in this case which are explicitly listed). In the caseRet=?1/2, an algebraic method is used, in the general case, Baker?s method yields the result.

A one-way function based on norm form equations

In this paper we present a new one-way function with collision resistance. The security of this function is based on the difficulty of solving a norm form equation. We prove that this function is collision resistant, so it can be used as a one-way hash function. We show that this construction probably provides a family of one-way functions.

Notes on CNS Polynomials and Integral Interpolation

Let P(x)=p d x d+⋯+p 0 ∈ ℤ[x], with p d =1. It is called a CNS polynomial if every element of the factor ring R = ℤ[x]/P(x)ℤ(x) has a unique representative of form

Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert

\sum\limits_{i = 0}^\ell {a_i x^i } , 0 \leqslant a_i < \left| {p_0 } \right|, 0 \leqslant i \leqslant \ell .

On the Diophantine equation Gn(x) = Gm(P(x)): Higher-order recurrences

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Reducible cubic CNS polynomials

In the present paper we study sequences defined by the recurrence relation for n ≥ 0, where the golden ratio. These sequences are related to shift radix systems as well as to β-expansions with respect to Salem numbers.

Elements with bounded height in number fields

A conjecture of W.J. Gilberts on canonical number systems which are defined by cubic polynomials is partially proved, and it is shown that the conjecture is not complete. Applications to power integral bases of simplest and pure cubic number fields are given thereby extending results of S. Kormendi.

A note on generalized radix representations and dynamical systems

Let be a zero of the Thomas polynomial X 3 (a 1)X 2 (a+2)X 1. We find all algebraic numbers µ = x0+x1 +x2 2 2 Z[ ], such that x0,x1,x2 2 Z forms an arithmetic progression and the norm of µ is less than |2a + 1|. In order to find all progressions we reduce our problem to solve a family of Thue equations and solve this family completely.