In this paper, we study the Diophantine equation bk+(a+ b)k+(2a+ b)k++(a(x-1) +b)k=y(y +c)(y +2c)(y +(& ell;-1) c) , where a, b, c, k, & ell; are given integers under natural conditions. We prove some effective results for special values for c, k and & ell;, and obtain a general ineffective result based on the Bilu-Tichy method.

In this paper we study the Diophantine equation \begin{align*} b^k + \left(a+b\right)^k +&\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) + b\right)^k = \\&y\left(y+c\right) \left(y+2c\right) \ldots \left(y+ \left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given integers under natural conditions. We prove some effective results for special values for $c,k$ and $\ell$ and obtain a general ineffective result based on Bilu-Tichy method.

In the present paper we prove that under certain conditions the linear combination of two Euler polynomials with odd degrees P n , m ( x ) = E n ( x ) + c E m ( x ) is always indecomposable over C , where c denotes a rational number.

In the present paper, we prove that the general integer linear combination of several Bernoulli and Euler polynomials with odd degree is always indecomposable over the field of complex numbers.

In this paper we determine possible decompositions of Euler polynomials $E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation $$-1^k +2 ^k - \cdots + (-1)^{x} x^k=g(y),$$ with $g\in \mathbb{Q}[X]$ of degree at least $2$ and $k\geq 7$, has only finitely many integers solutions $x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.

In this note is proved that there is no or at most one non-zero complex number b such that the shifted Bernoulli polynomial Bk(x)+b has no at least three zeros of odd multiplicities for odd or even values of k⩾7, respectively.

A kutatocsoport tagjai jelentős eredmenyeket ertek el a szamelmelet, es ezen belul a diofantikus egyenletek elmeleteben. Effektiv es ineffektiv vegessegi teteleket nyertek toruszok bizonyos reszvarietasainak pontjaival kapcsolatban, es fuggvenytestek illetve szamtestek feletti rezultans forma egyenletek megoldasaira. Teljesen megoldottak Thue- illetve szuperelliptikus egyenletcsaladokat es kulonboző exponencialis diofantikus egyenleteket. Uj eredmenyeket nyertek veges alaptestű fuggvenytestek feletti diofantikus egyenletekről, folytattak kutatasaikat algebrai szamtestek hatvany egesz bazisaival kapcsolatban. Klasszikus teteleket altalanositva, vizsgaltak a szamtani sorozatokban előfordulo teljes hatvanyokat. Uj eredmenyeket nyertek az alkalmazasok szempontjabol fontos szomszedsagi szekvenciak elmeleteben, valamint a diszkret tomografiaban. Altalanositottak a balansz szamok fogalmat, ineffektiv vegessegi allitasokat bizonyitottak kulonboző szeparabilis diofantikus egyenletek megoldasszamara. Vizsgaltak index formak kriptografiai felhasznalhatosagat. Leirtak kulonboző polinomcsaladok illetve eltoltjaik gyokszerkezetet. Reszben a fenti eredmenyeket felhasznalva, Rakaczki Csaba, Pink Istvan es Nyul Gabor megszerezte a PhD fokozatot, Berczes Attila es Pinter Akos elkeszitette habilitacios illetve MTA doktori ertekezeset. | The members of the research group have obtained significant results in number theory, in particular concerning Diophantine equations. Effective and ineffective theorems have been derived about points of certain subvarieties of tori, and also for the solutions of resultant form equations over number fields and function fields. Families of Thue- and superelliptic equations, as well as several exponential Diophantine equations have been resolved. New results for Diophantine equations over finite fields have been obtained, and the research about power integral bases of algebraic number fields has also been continued. Generalizing classical theorems, perfect powers in arithmetic progressions have been investigated. New results have been proved in the theory of neighborhood sequences and in discrete tomography, which may have significant applications later on. The notion of balancing numbers has been generalized, and ineffective finiteness results have been derived for the number of solutions of several separable Diophantine equations. The cryptographical applicability of index forms has been investigated. The root structures of several families of polynomials and their translations have been described. Partly based upon the above results, Csaba Rakaczki, Istvan Pink and Gabor Nyul have received a PhD degree, and Attila Berczes and Akos Pinter has submitted a habilitation thesis and an Academical Doctoral dissertation, respectively.

In this paper we prove that there is at most one complex number b for which the shifted Euler polynomial E n (x) + b has at most two zeros of odd multiplicity.

Szamos jelentős effektiv, kvantitativ es numerikus eredmeny szuletett egy sor alapvető fontossagu diofantikus problemaval kapcsolatban. Az eredmenyek elsősorban szeteső forma egyenletekre, S-egysegegyenletekre, szuperelliptikus es binom Thue egyenletekre, altalanositott Fermat-tipusu egyenletekre, valamint rekurziv sorozatokra, adott diszkriminansu, illetve adott rezultansu polinomokra es biner formakra, altalanositott szamrendszerekre es alkalmzasaikra vonatkoznak. A legkiemelkedőbb eredmenyek a kovetkezők. Teljesen explicit eredmenyt nyertek a hires ABC-sejtes szamtestek feletti altalanositott valtozataval kapcsolatban. 13-nal nagyobb kitevők eseten megoldottak a Fermat-fele egyenlet bizonyos fontos altalanositasait. Kozos altalanositasat adtak az ismeretlen fokszamu binom Thue egyenletekre es az S-egysegegyenletekre vonatkozo korabbi nevezetes (kvalitativ) effektiv vegessegi teteleknek. Jelentős attorest hajtottak vegre egy tobb evszazados problemakorben, megmutatvan, hogy legfeljebb 11 tagu szamtani sorozat tagjainak a szorzata (bizonyos trivialis kivetelektől eletekintve) nem lehet teljes hatvany. Uj modszereket, hatekony eljarasokat dolgozatk ki ismeretlen fokszamu binom Thue egyenletek, S-egysegegyenletek, szuperelliptikus egyenletek, altalanositott Fermat-fele egyenletek, valamint index forma egyenletek konkret esetekben valo megoldasara. Mindezeknek szamos fontos alkalmazasat adtak a diofantikus szamelmeletben es az algebari szamelmeletben. | Several effective, quantitative and numerical results have been established on various diophantine problems of fundamental importance. These results concern mostly decomposable form equations, S-unit equations, superelliptic equations, binomial Thue equations, generalized Fermat-type equations, linear recurrences, binary forms of given discriminant resp. of given resultant, generalized number systems and their applications. The most important scientific achievements of the project are as follows. A completely explicit result has been obtained in the direction of the famous ABC-conjecture for number fields. Certain important generalizations of the Fermat equation have been solved for exponents greater than 13. A common generalization has been given of the earlier (qualitative) effective finiteness theorems concerning S-unit equations resp. binomial Thue equations with unknown exponent. A considerable breakthrough has been made in connection with a problem going back to Fermat and Euler: it has been proved that (apart from some trivial exceptions) a product of at most 11 consecutive terms in an arithmetic progression can never be a perfect power. New methods and efficient algorithms have been elaborated for solving, in concrete cases, binomial Thue equations with unknown exponent, S-unit equations, superelliptic equations, generalized Fermat-type equations and index form equations. These led to many important applications in diophantine and algebraic number theory.

As a generalization of the results of [3] and [22], we characterize those pairs (m,n) and those polynomials b{Z}[x] of prime degree for which equation (1) has only finitely many integer solutions.

In this paper we characterize those polynomials g(y) with rational coefficients and positive integers m for which 1(m) + 2(m) + (...) + x(m =) g(y) has infinitely many integer solutions. As an application of this result we give an ineffective finiteness result concerning equation S-m(x) = F((y/n)), where F(y) is an element of phi\y\.