On a simple quartic family of Thue equations over imaginary quadratic number fields

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality $$|F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1$$ has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.Comment: 27 page

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Thue's Fundamentaltheorem, I: The General Case

In this paper, Thue's Fundamentaltheorem is analysed. We show that it includes, and often strengthens, known effective irrationality measures obtained via the so-called hypergeometric method as well as showing that it can be applied to previously unconsidered families of algebraic numbers. Furthermore, we extend the method to also cover approximation by algebraic numbers in imaginary quadratic number fields.Comment: accepted version (Acta Arithmetica

On a parametric family of Thue inequalities over function fields

In this paper we completely solve a family of Thue inequalities defined over the field of functions $\mathbb{C}(T)$, namely deg (X4−4cX3Y+(6c+2)X2Y2+4cXY3+Y4) ≤ deg c, where the solutions x,y come from the ring $\mathbb{C}\left[ T\right]$ and the parameter $c\in \mathbb{C}\left[ T\right]$ is some non-constant polynomia

Effective Methods for Norm-Form Equations

While effective resolution of Thue equations has been well understood since the work of Baker in the 1960s, similar results for norm-form equations in more than two variables have proven difficult to achieve. In 1983, Vojta was able to address the case of three variables over totally complex and Galois number fields. In this paper, we extend his results to effectively resolve several new classes of norm-form equations. In particular, we completely and effectively settle the question of norm-form equations over totally complex Galois sextic fields.Comment: Final version, accepted by Math Annalen. A few changes from the previous version-- in particular there is a new result that also applies over non-Galois extensions. The explicit example was removed and will appear elsewher

Algoritmikus számelmélet és alkalmazásai a kriptográfiában = Computational number theory and its applications in cryptography

Megmutattuk, hogy a normaforma függvényt moduló pq redukálva, ahol p és q nagy prímszámok, ütközésmentes függvényt kapunk. Több konstrukciót elemeztünk kriptográfiai alkalmazások szempontjából releváns véletlen sorozatcsaládra. Ha Gn(x) egy algebrailag zárt test feletti lineáris rekurzív polinomsorozat és x,y algebrailag függőek, akkor bebizonyítottuk, hogy a Gn(x)=Gm(y) egyenletnek általános feltételek mellett csak véges sok megoldása van n,m-ben. Bebizonyítottuk, hogy egy normaforma egyenletnek általában csak véges sok olyan megoldása van, ahol a megoldások koordinátái egy számtani sorozatot alkotnak. Megadtuk a klasszikus ?-reprezentáció és a kanonikus számrendszerek egy közös általánosítását és több dolgozatban vizsgáltuk ezen SRS-nek elnevezett fogalom tulajdonságait. Elemzésünk választ ad arra, hogy miért nehéz a harmadfokú CNS polinomok, illetve az (F) tulajdonságú Pisot számok jellemzése. Megfogalmaztuk a következő sejtést: Legyen |?|<2 és {an} egész számok olyan sorozata, amelyre 0 ? an-1+ ? an + an+1 <1 minden n-re. Akkor {an} periódikus. Számos korábbi eredményt messzemenően általánosítva, mély eredményeket értünk el két klasszikus, Fermat-ig és Eulerig visszanyúló diofantikus témakörben, nevezetesen számtani sorozatokban, illetve hatványösszegekben található teljes hatványokra vonatkozóan. Többek között megmutattuk, hogy egészekből álló k-tagú számtani sorozat tagjainak szorzata k ? 11-re általában nem lehet teljes hatvány. | Considering the reduction modulo pq, where p and q are big primes we constructed collision resistant hash functions. We studied some construction of cryptography relevant pseudo random number sequences. If Gn(x) denotes a linear recursive polynomial sequence over an algebraically closed field and x,y are algebraically dependent, then we proved that the equation Gn(x)=Gm(y) has under quite general assumptions only finitely many solutions in n,m. We proved that a norm form equation has only finitely many solutions, which coordinates form an arithmetical progression. We realized a common generalization, called shift radix system, of the classical ?-reprezentation and the canonical number systems and studied its properties in several papers. Our investigation showed that the characterization problem of cubic CNS polynomials and Pisot numbers of proprty (F) is complicated. We made rise the conjecture: Let |?|<2 and {an} a sequence of integers staisfying the inequality 0 ? an-1+ ? an + an+1 <1 for all n. Then {an} is periodical. Generalyzing essentially several earlier results, we achieved deep results in two classical diophantine topics: perfect powers in arithmetical progressions and in power sums, which are going back to Farmat and Euler. We proved among others that the product of members of an arithmetical progression of length at most 11 apart from trivial cases cannot be a perfect power