Effective, quantitative and computational investigations in Diophantine number theory  Page description

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Details of project

 
Identifier
100339
Type K
Principal investigator Győry, Kálmán
Title in Hungarian Effektív, kvantitatív és számítógépes vizsgálatok a diofantikus számelméletben
Title in English Effective, quantitative and computational investigations in Diophantine number theory
Keywords in Hungarian diofantikus egyenletek
Keywords in English Diophantine equations
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Number theory
Panel Mathematics and Computing Science
Department or equivalent Department of Algebra and Number Theory (University of Debrecen)
Participants Bérczes, Attila
Gaál, István
Hajdu, Lajos
Kovács, Tünde
Nyul, Gábor
Pethő, Attila
Pintér, Ákos
Tengely, Szabolcs
Starting date 2012-01-01
Closing date 2016-06-30
Funding (in million HUF) 15.704
FTE (full time equivalent) 11.70
state closed project
Summary in Hungarian
A debreceni számelméleti iskola az elmúlt 35 évben nemzetközi viszonylatban meghatározó szerepet játszott a diofantikus számelméletben. A modern kutatások szinte valamennyi fő irányában élvonalbeli, esetenként áttörést jelentő eredményeket értünk el. A jelen pályázat keretein belül folytatni kívánjuk korábbi vizsgálatainkat. Emellett új irányokban is tervezünk kutatásokat. Kutatási elképzeléseinket a következő témakörök köré csoportosíthatjuk:

1) Minél általánosabb végességi kritériumok feltárása; végtelen sok megoldás esetén a megoldások eloszlásának kvantitatív vizsgálata, véges sok megoldás esetén a lehető legélesebb korlátok levezetése a megoldásszámra, valamint magukra a megoldásokra (széteső polinom egyenletek, egységegyenletek és közös általánosításaik, szuperelliptikus egyenletek).

2) Hatékony algoritmusok kidolgozása S-egész illetve S-egység megoldások meghatározására (index forma, S-egység- és elliptikus egyenletek).

3) Az összes megoldás meghatározása Fermat-típusú és binom Thue egyenletek bizonyos parametrikus családjai esetén.

4) A nyert eredmények alkalmazásai az algebrai számelméletben, a rekurzív sorozatok elméletében, valamint számtani sorozatok tagjainak szorzatában található teljes hatványok meghatározására.

A vizsgálatok során a jelenlegi modern, mély módszerek (pl. Baker-módszer, Wiles-módszer, számítógépes eljárások) alkalmazásán kívül szükségünk lesz azok bizonyos továbbfejlesztésére, új módszerek kidolgozására és mindezek alkalmas kombinálására.

Kutatásainkat egymással, valamint fiatal debreceni munkatársainkkal, hazai és külföldi kollégákkal együttműködve folytatjuk.
Summary
In the last three decades the number theory research group in Debrecen played an improtant, internationally recognized role in diophantine number theory. We have achieved significant results in almost all main research areas, making a breakthrough in some cases. In the frame of the present project we intend to continue our previous research activity. Beside this, we plan to start research into new directions as well. We intend to concentrate on the following research topics:

1) Establishment of new finiteness criteria of much general type; in case of infinitely many solutions the quantitative investigation of the set of solutions, in case of finitely many solutions deriving sharp bounds for the number of solutions and for the solutions themselves (decomposable polynomial equations, unit equations and their common generalizations, superelliptic equations).

2) Development of efficient algorithms for determining the S-integer solutions and S-unit solutions (index form equations, S-unit equations and elliptic equations).

3) Determination of all solutions in case of certain parametric families of Fermat-type equations and binom Thue-equations.

4) Applications of our results in algebraic number theory, in the theory of recurrence sequences, and to the description of perfect powers in products of consecutive terms of arithmetic progressions.

In these investigations, beside the use of the present modern, deep methods (such as Baker’s method, Wiles’ method and computational procedures) we shall need certain improvements of these methods and developments of new ones, as well as the combinations of them.

We shall do this research in collaboration with each other, and also with colleagues from abroad.





 

Final report

 
Results in Hungarian
Számos jelentős effektív, kvantitatív és explicit eredmény született egy sor alapvető fontosságú diofantikus problémával kapcsolatban. Az eredmények elsősorban egységegyenletekre és általánosításaikra, diszkrimináns egyenletekre, Thue egyenletekre és általánosításaikra, szuperelliptikus egyenletekre, hatványösszeg típusú egyenletekre és alkalmazásaikra vonatkoznak. A legkiemelkedőbb eredmények a következők. Az egységegyenletek és a diszkrimináns egyenletek centrális szerepet játszanak a diofantikus számelméletben. GYŐRY, Evertsével közösen, az első átfogó munkákat publikálta a Cambridge University Pressnél egységegyenletekről illetve diszkrimináns egyenletekről és alkalmazásaikról. A könyvek a szerzők számos korábbi és sok új eredményét is tartalmazzák. Továbbá a szerzők módszert dolgoztak ki a számtestek feletti diofantikus egyenletek effektív elméletének a végesen generált esetre való kiterjesztésére. BÉRCZES effektivizálta Lang, valamint Liardet általánosított egységegyenletekre vonatkozó régi, nevezetes tételeit végesen generált tartományok felett. HAJDU, Bertókkal egy természetes sejtés feltételezése mellett hatékony eljárást dolgozott ki konkrét többismeretlenes S-egységegyenletek megoldására. GYŐRY és HAJDU, Tijdemannal közösen, több alapvető eredményt nyert S-egységek különbség gráfjainak reprezentációiról.
Results in English
Several effective, quantitative and explicit results have been established on various diophantine problems of fundamental importance. These results concern mostly unit equations and their generalizations, discriminant equations, Thue equations and their generalizations, superelliptic equations, equations of power sum type and their applications. The most important scientific achievements of the project are as follows. Unit equations and discriminant equations play a central role in diophantine number theory. GYŐRY, together with Evertse, gave the first comprehensive accounts about these equations and their various applications. Their books, published at the Cambridge University Press in 2015 resp. 2016, contain a great number of earlier as well as new results of the authors. Further, the authors developed a general method to extend the effective theory of diophantine equations over number fields to the case of finitely generated ground domains. BÉRCZES made effective some remarkable theorems of Lang and Liardet on generalized unit equations over finitely generated domains. Under a natural conjecture, HAJDU and Bertók worked out an efficient algorithm for solving concrete S-unit equations in several unknowns. GYŐRY, HAJDU and Tijdeman established some results of basic importance on the representations of difference graphs of S-units.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=100339
Decision
Yes





 

List of publications

 
A. PETHŐ, P. Varga,: Canonical number systems over imaginary quadratic Euclidean domains,, Colloq. Math. 146, 165-186, 2017
I. GAÁL, M. Pohst,: The sum of two S-unit being a perfect power in global function fields,, Mathematica Slovaca, 63, 69-76,, 2013
SZ. TENGELY, M. Ulas,: On products of disjoint blocks of arithmetic progressions and related equations,, J. Number Theory, 165, 67-83,, 2016
Á. PINTÉR and Cs. Rakaczki,: On the decomposability of linear combinations of Bernoulli polynomials,, Monatshefte für Mathematik, 180(3), 631-648,, 2016
I. GAÁL,: Calculating "Small" Solutions of Relative Thue Equations,, Experimental Mathematics, 24, 142-149,, 2015
J. H. Evertse and K. GYŐRY: Unit Equations in Diophantine Number Theory,, Cambridge University Press, megjelenés alatt, 2014
A. Dujella, K. GYŐRY and Á. PINTÉR: On power values of pyramidal numbers, I,, Acta Arith., 155, 217-226, 2012
J. H. Evertse and K. GYŐRY,: Effective results for unit equations over finitely generated domains,, Math. Proc. Cambridge Philos. Soc. 154, 351-380, 2013
A. BÉRCZES, J.H. Evertse and K. GYŐRY,: Multiply monogenic orders, Ann Scuola Normale Sup. Pisa, 12, 467-497, 2013
A. BÉRCZES, J.H. Evertse and K. GYŐRY,: Effective results for hyper- and superelliptic equations over number fields,, Publ. Math. Debrecen, 82, 727-756,, 2013
D. Dombek, L. HAJDU and A. PETHŐ,: Representing algebraic integers as linear combinations of units, Periodica Mathematica Hungarica, 68, 135-142,, 2014
S. Akiyama and A. PETHŐ,: Discretized rotation has infinitely many periodic orbits, Nonlinearity, 26, 871-880, 2013
S. Akiyama and A. PETHŐ,: On the distribution of polynomials with bounded roots I. Polynomials with real coefficients., J. Math. Soc. Japan, 66, 927-949,, 2014
C. Fieker, I. GAÁL, and M. Pohst,: On computing integral points of a Mordell curve over rational function fields in characteristic > 3,, J. Number Theory 133, 738-750., 2013
I. GAÁL and G. Petrányi,: Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields,, Czech. Math. J., 64, 465-475,, 2014
A. Bazsó, Á. PINTÉR and H. Srivastava,: A refinement of Faulhaber's theorem concerning sums of powers of natural numbers,, Applied Mathematics Letters, 25. 486-489., 2012
Á. PINTÉR and V. Ziegler,: On arithmetic progressions in recurrences - A new characterization of the Fibonacci sequence,, Journal of Number Theory, 132, 1686-1706., 2012
Á. PINTÉR and H. Srivastava,: Addition theorems for the Appel polynomials and the associated classes of polynomial expansions, Aequationes Mathematicae, 85, 483-495,, 2013
A. Bazsó, D. Kreso, F. Luca and Á. PINTÉR,: On equal values of power sums of arithmetic progressions,, Glasnik Math., 47, 253-263., 2012
Y. Bilu, C. Fuchs, F. Luca and Á. PINTÉR,: Combinatorial diophantine equations and a refinement of a theorem on separated variables equations,, Publ. Math. Debrecen, 82. 219-254., 2013
Á. PINTÉR and SZ. TENGELY,: The Korteweg-de Vries equation and a diophantine problem related to Bernoulli polynomials,, Advances in Difference Equations, 2013: 245, 2013
L. HAJDU and M. Szikszai,: On the GCD-s of k consecutive terms of Lucas sequences,, J. Number Theory 132. 3056-3069., 2012
L. HAJDU and R. Tijdeman,: Representing integers as linear combinations of power products,, Arch. Math. 98, 527-533., 2012
L. HAJDU and V. Ziegler,: Distinct unit generated totally complex quartic fields,, Math. Comp. 83, 1495-1512,, 2014
B. van Dalen, L. HAJDU és R. Tijdeman,: Bounds for discrete tomography solutions, Indag. Math. 24, 391-402;, 2013
L. HAJDU and M. Szikszai,: On the GCD-s of k consecutive terms of an elliptic divisibility sequence, Publ-Math. Debrecen, 84, 291-301,, 2014
A. BÉRCZES, A. Dujella and L. HAJDU,: Some diophantine properties of the sequence of S-units, J. Number Theory, 138, 48-68,, 2014
L. HAJDU and R. Tijdeman,: Bounds for approximate discrete tomography solutions, SIAM Journal on Discrete Mathematics, 27, 1055-1066,, 2013
L. Kovács, H. Tomán, Á. Jónás L. HAJDU and A. Hajdu,: Generalizing the majority voting scheme to spatially constrained voting,, IEEE Transactions on Image Processing, 22, 4182-4194,, 2013
A. BÉRCZES and F. Luca,: On the sum of digits of numerators of Bernoulli numbers,, Canad. Math. Bull. 56, 723-728, 2013
SZ. TENGELY,: Balancing numbers which are products of consecutive integers,, Publ. Math. Debrecen, 83, 197-205;, 2013
C. Fuchs, A. PETHŐ, SZ. TENGELY: On decomposable rational functions with given number of singularities,, Algebraic Systems and Theoretical Computer science, Kyoto Univ. Res. Inst. for Math.1809, 54-64,, 2012
Zs. Kereskényi-Balogh, and G. NYUL,: Stirling numbers of the second kind and Bell numbers for graphs,, Australian J. of Combinatorics, 58, 264-274,, 2014
Zs. Rábai and T. KOVÁCS,: On equal values of pyramidal numbers,, Indag. Math., közlésre elfogadva, 2013
A. BÉRCZES, J. H. Evertse and K. GYŐRY,: Effective results for Diophantine equations over finitely generated domains,, Acta Arith., 163, 71-100,, 2014
K. GYŐRY, T. KOVÁCS, Gy. Péter and Á. PINTÉR,: Equal values of standard counting polynomials,, Publ. Math. Debrecen, 84, 259-277, 2014
V. Komornik and A. PETHŐ,: Common expansions noninteger bases,, Publ. Math. Debrecen, 85, 489-501,, 2014
B. He, I. Pink, Á. PINTÉR and A. Togbé,: On the diophantine inequality |x2-cxy2 + y4| ≤ c+2,, Glasnik Matematicki, 48, 291-299;, 2013
K. J. Batenburg, W. Fortes, L. HAJDU and R. Tijdemann,: Bounds on the quality of reconstructed images in binary tomography,, Disc. Appl. Math. 161, 2236-2251,, 2013
L. HAJDU, Á. PINTÉR, SZ. TENGELY and N. Varga,: Equal values of figurate numbers,, J. Number Theory, 137, 130-141,, 2014
S. Brunetti, P. Dulio, L. HAJDU, C. Peri,: Ghosts in Discrete Tomography, Journal of Mathematical Imaging and Visions, 53, 210-224,, 2015
A. BÉRCZES and V. Ziegler,: On geometric progressions on Pell equations and Lucas sequences,, Glasnik Matematicki, 48: 1-22,, 2013
A. BÉRCZES and A. PETHŐ,: On the sumset of binary recurrences sequences,, Publ. Math. Debrecen, 84, 279-290, 2014
G. NYUL and G. Rácz,: The r-Lah numbers,, Discrete Mathematics, 338, 1660-1666, 2015
G. NYUL,: Pexiderized functional equations for vector products and quaternions,, Acta Math. Hungar., 142, 519-525,, 2014
M. A. Bennett and Á. PINTÉR,: Intersections of recurrence sequences,, Proc. Amer. Math. Soc., 143, 2347-2353,, 2015
Bo He, Á. PINTÉR, A. Togbé, N. Varga,: A generalization of a problem of Mordell, Glasnik Matematicki, 50, 35-41,, 2015
Bo He, Á. PINTÉR, A. Togbé,: On simultaneous Pell equation and related Thue equations, Proc. Amer. Math. Soc., 143, 4685-4693, 2015
L. HAJDU and I. Pink,: On the Diophantine equation 1+2^a+x^b=y^n,, J. Number Theory, 143, 1-13,, 2014
K. GYŐRY, L. HAJDU, R. Tijdeman,: Representation of finite graphs as difference graphs of S-units,, I, J. Combin. Th. Series A, 127, 314-335,, 2014
A. BÉRCZES, L. HAJDU, N. Hirata-Kohno, T. KOVÁCS, A. PETHŐ,: A key exchange protocol based on Diophantine equations and S-integers,, JSIAM Letters, 6, 85-88,, 2014
Cs. Bertók, L. HAJDU,: A Hasse-type principle for exponential diophantine equations and its applications,, Math. Comp., 85, 849-860,, 2016
L. HAJDU,: On a conjecture of Schaffer concerning the equation S_k(x)=y^n, J. Number Theory, 155, 129-138,, 2015
L. HAJDU és M. Szikszai.,: Common factors in series of consecutive terms of associated Lucas and Lehmer sequences, Fibonacci Quart, 53, 221-229,, 2015
A. Custic, L. HAJDU, D. Kreso, R. Tijdeman,: On conjectures and problems of Ruzsa, concerning difference graphs of S-units, Acta Math, Hungar, 164, 399-404,, 2015
S. Akiyama, L. Aszalós, L. HAJDU, A. PETHŐ,: Correlation clustering of graphs and integers, Infocommunications J., 6, 3-12, 2014
A. BÉRCZES: Effective results for unit points on curves over finitely generated domains,, Math. Proc. Cambridge Phil. Soc., 158, 331-353,, 2015
SZ. TENGELY, N. Varga,: On a generalization of a problem of Erdős and Graham,, Publ. Math. Debrecen, 84, 475-482,, 2014
A. BÉRCZES, A. Dujella, L. HAJDU, SZ. TENGELY,: Finiteness results for F-Diophantine sets,, Monatshefte für Mathematik, 180, 469-484,, 2016
G. NYUL and B. Rauf,: On the existence of van der Waerden type numbers for linear recurrence sequences with constant coefficients,, Fibonacci Quarterly, 53, 53-60,, 2015
J. H. Evertse and K. GYŐRY: Unit Equations in Diophantine Number Theory, Cambridge University Press,, 2015
A. Bazsó, A. BÉRCZES, K. GYŐRY, Á. PINTÉR,: Erratum to the paper "On the resolution of equations Axn-Byn=C in integers x, y and n≥3, II, Publ. Math. Debrecen, 76, 227-250, 2010,, Publ. Math. Debrecen, 86, 251-252,, 2015
J. H. Evertse, K. GYŐRY,: Discriminant Equations in Diophantine Number Theory,, Cambridge University Press,, 2017
K. GYŐRY, L. HAJDU, R. Tijdeman,: Representation of finite graphs as difference graphs of S-units, II,, Acta Math. Hungar. 149, 423-447,, 2016
J. H. Evertse, K. GYŐRY,: Effective results for discriminant equations over finitely generated integral domains,, Number Theory-Diophantine problems uniform distribution and applications, megjelenés alatt, 2017
A. PETHŐ, M. Pohst, Cs. Bertók,: On multidimensional Diophantine approximation of algebraic numbers,, J. Number Theory, 171, 422-448,, 2017
L. Aszalós, L. HAJDU, A. PETHŐ,: On a correlational clustering of integers,, Indag. Math. (N.S.) 27, 173-191,, 2016
Cs. Bertók, L. HAJDU, A. PETHŐ,: On the distribution of polynomials with bounded height,, J. Number theory, közlésre benyújtva, 2017
V. Komornik, M. Pedicini, A. PETHŐ,: Multiple common expansions in nonnegative integer bases,, Acta Sci. Math. (Szeged), megjelenés alatt, 2017
I. GAÁL, L. Remete,: Non-monogenity in a family of octic fields,, Rocky Mountain J. Math., megjelenés alatt, 87-100, 2017
L. HAJDU, S. Laishram, SZ. TENGELY,: Power values of sums of products of consecutive integers,, Acta Arith. 172, 333-349,, 2016
A. BÉRCZES, L. HAJDU, T. Miyazaki, I. Pink,: On the equation 1^k+2^k+…+x^k=y^n for fixed x,, J. Number Theory, 163, 43-60, 2016
L. HAJDU, N. Saradha,: On generalizations of problems of Recaman and Pomerance,, J. Number Theory 162, 552-563,, 2016
L. HAJDU, S. Laishram, M. Szikszai,: Perfect powers in products of terms of elliptic divisibility sequences,, Bull. Austral. Math. Soc. 94, 395-404,, 2016
A. BÉRCZES, L. HAJDU, T. Miyazaki, I. Pink,: On the Diophantine equation 1+x^a+z^b=y^n,, Journal of Combinatorics and Number Theory 8, 145-154,, 2016
C. Fuchs, L. HAJDU,: 30 years of cooperation,, Period. Math. Hung. közlésre elfogadva., 2017
Cs. Bertók, L. HAJDU, I. Pink, Zs. Rábai,: Linear combinations of prime powers in binary recurrence sequences,, International Journal of Number Theory közlésre elfogadva, 2017
Cs. Bertók, L. HAJDU, F. Luca, D. Sharma,: On the number of non-zero digits of integers in multi-base representations,, Publ. Math. Debrecen, 90, 181-194,, 2017
A. Hajdu, B. Harangi, R. Besenczi, I. Lazar, G. Emri, L. HAJDU, R. Tijdeman,: Measuring Regularity of Network Patterns by Grid Approximations using the LLL Algorithm, 23rd, International Conference on Pattern Recognition (ICPR 2016), Cancun, Mexico, közlésre elfogadva, 2016
A. Hajdu, H. Toman, L. Kovács, L. HAJDU,: Composing ensembles of object detectors under execution time constraint, 23rd International Conference on Pattern Recognition (ICPR 2016), Cancun, Mexico, 2016, közlésre elfogadva, 2016
A. Hajdu, L. HAJDU, R. Tijdeman,: Finding well approximating lattices for a finite set of points,, Mathematics of Computations, közlésre benyújtva, 2017
L. HAJDU, R. Tijdeman,: Consistency conditions for discrete tomography,, Fundamenta Informaticae, közlésre benyújtva, 2017
G. NYUL,: Some thoughts concerning power sums,, Teaching Mathematics and Computer Science, 13, 303-308,, 2015
Cs. Bertók, G. NYUL,: On monochromatic linear recurrence sequences,, Contributions to Discrete Mathematics, közlésre elfogadva, 2017
BÉRCZES A, Ziegler V,: On simultaneous palindromes,, Journal of Combinatorics and Number Theory, 6, 37-49,, 2014
BÉRCZES A.: Effective results for division points on curves in G_m^2,, J. Théor. Nombres Bordeaux, 27: 405-437.., 2015
BÉRCZES A, Luca F, Pink I, Ziegler V.: Finiteness results for Diophantine triples with repdigit values,, Acta Arith., 172, 133–148., 2016
BÉRCZES A.: Effective results for Diophantine problems over finitely generated domains,, MTA doktori disszertáció,, 2016
K. GYŐRY: Perfect powers in products with consecutive terms from arithmetic progressions, II,, Erdős Centennial, Springer, 311-324;, 2013
K. GYŐRY and Á. PINTÉR: Binomial Thue equations, ternary equations and power values of polynomials, J. Math. Sciences 180, 569-580., 2012
A. Dujella, K. GYŐRY and Á. PINTÉR: On power values of pyramidal numbers, I,, Acta Arith., 155, 217-226, 2012
J.H. Evertse and K. GYŐRY,: Effective results for unit equations over finitely generated domains,, Math. Proc. Cambridge Philos. Soc. 154, 351-380, 2013
A. BÉRCZES, J.H. Evertse and K. GYŐRY,: Multiply monogenic orders, Ann Scuola Normale Sup. Pisa, 12, 467-497, 2013
A. BÉRCZES, J.H. Evertse and K. GYŐRY,: Effective results for hyper- and superelliptic equations over number fields,, Publ. Math. Debrecen, 82, 727-756,, 2013
D. Dombek, L. HAJDU and A. PETHŐ,: Representing algebraic integers as linear combinations of units, Periodica Mathematica Hungarica, 2013
S. Akiyama and A. PETHŐ,: Discretized rotation has infinitely many periodic orbits, közlésre benyújtva, 2013
S. Akiyama and A. PETHŐ,: On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients, közlésre benyújtva,, 2013
S. Akiyama and A. PETHŐ,: On the distribution of polynomials with bounded roots I. Polynomials with real coefficients., J. Math. Soc. Japán, közlésre elfogadva, 2013
L. HAJDU, T. KOVÁCS, A. PETHŐ and M. Pohst: An optimization problem for lattices, Experimental Math., 22, 443-455;, 2013
I. GAÁL and T. Szabó,: Power integral bases in parametric families of biquadratic fields,, JP Journal of Algebra, Number Theory and Applications, 21, 105-114., 2012
C. Fieker, I. GAÁL, and M. Pohst,: On computing integral points of a Mordell curve over rational function fields in characteristic > 3,, J. Number Theory 133, 738-750., 2013
I. GAÁL and G. Petrányi,: Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields,, közlésre benyújtva, 2013
I. GAÁL and T. Szabó,: Relative power integral bases in infinite families of quartic extensions of quadratic field,, JP Journal of Algebra, Number Theory and Applications, 29, 31-43;, 2013
A. Bazsó, Á. PINTÉR and H. Srivastava,: A refinement of Faulhaber's theorem concerning sums of powers of natural numbers,, Applied Mathematics Letters, 25. 486-489., 2012
Á. PINTÉR and V. Ziegler,: On arithmetic progressions in recurrences - A new characterization of the Fibonacci sequence,, Journal of Number Theory, 132, 1686-1706., 2012
Á. PINTÉR and H. Srivastava,: Addition theorems for the Appel polynomials and the associated classes of polynomial expansions, Aequationes Mathematicae, 85, 483-495,, 2013
A. Bazsó, D. Kreso, F. Luca and Á. PINTÉR,: On equal values of power sums of arithmetic progressions,, Glasnik Math., 47, 253-263., 2012
Y. Bilu, C. Fuchs, F. Luca and Á. PINTÉR,: Combinatorial diophantine equations and a refinement of a theorem on separated variables equations,, Publ. Math. Debrecen, 82. 219-254., 2013
D. Kim, Y. K. Park and Á. PINTÉR,: A diophantine problem concerning polygonal numbers,, Bulletin of Australian Math Society, 88, 345-350;, 2013
Á. PINTÉR and SZ. TENGELY,: The Korteweg-de Vries equation and a diophantine problem related to Bernoulli polynomials,, Advances in Difference Equations, 2013: 245, 2013
L. HAJDU and M. Szikszai,: On the GCD-s of k consecutive terms of Lucas sequences,, J. Number Theory 132. 3056-3069., 2012
L. HAJDU and R. Tijdeman,: Representing integers as linear combinations of power products,, Arch. Math. 98, 527-533., 2012
L. HAJDU and V. Ziegler,: Distinct unit generated totally complex quartic fields,, Math. Comp., 2013
B. van Dalen, L. HAJDU és R. Tijdeman,: Bounds for discrete tomography solutions, Indag. Math. 24, 391-402;, 2013
L. HAJDU and M. Szikszai,: On common factors within a series of consecutive terms of an elliptic divisibility sequence, Publ-Math. Debrecen, közlésre elfogadva, 2013
A. BÉRCZES, A. Dujella and L. HAJDU,: Some diophantine properties of The sequence of S-units, J. Number Theory, 2013
L. HAJDU and R. Tijdeman,: Bounds for approximate discrete tomography solutions, SIAM Journal on Discrete Mathematics, közlésre elfogadva, 2013
L. Kovács, H. Tomán, Á. Jónás L. HAJDU and A. Hajdu,: Generalizing the majority voting scheme to spatially constrained voting,, IEEE Transactions on Image Processing, közlésre elfogadva, 2013
A. BÉRCZES and F. Luca,: On the sum of digits of numerators of Bernoulli numbers,, Canad. Math. Bull. 56, 723-728, 2013
SZ. TENGELY,: Balancing numbers which are products of consecutive integers,, Publ. Math. Debrecen, 83, 197-205;, 2013
C. Fuchs, A. PETHŐ, SZ. TENGELY: On decomposable rational functions with given number of singularities,, Algebraic Systems and Theoretical Computer science, Kyoto Univ. Res. Inst. for Math., 2012
G. NYUL, and Zs. Balogh,: Stirling numbers of the second kind and Bell numbers for graphs,, Australian J. of Combinatorics, megjelenés alatt, 2013
G. NYUL and E. Gyimesi,: A note on Golomb's method and the continued fraction method for Egyptian fractions, Annales Mathematicae et Informaticae, megjelenés alatt,, 2013
Zs. Rábai and T. KOVÁCS,: On equal values of pyramidal numbers,, közlésre előkészítve, 2013
J. H. Evertse and K. GYŐRY,: Effective results for Diophantine equations over finitely generated domains: A survey, Turan Memorial Volume,, Number Theory, Analysis and Combinatorics, de Gruyter, 2013, 63-73., 2013
A. BÉRCZES, J. H. Evertse and K. GYŐRY,: Effective results for Diophantine equations over finitely generated domains,, Acta Arith.,, 2013
K. GYŐRY, T. KOVÁCS, Gy. Péter and Á. PINTÉR,: Equal values of standard counting polynomials,, Publ. Math. Debrecen,, 2013
K. GYŐRY, L. HAJDU and R. Tijdemann,: Representation of finite graphs as difference graphs of S-unis, I,, J. Combinatorial Theory,, 2013
A. PETHŐ and SZ. TENGELY,: On composite rational functions, Turan Memorial Volume,, Number Theory, Analysis and Combinatorics de Gruyter, 241-260,, 2013
V. Komornik and A. PETHŐ,: Common expansions noninteger bases,, közlésre benyújtva, 2013
B. He, I. Pink, Á. PINTÉR and A. Togbé,: On the diophantine inequality |x2-cxy2 + y4| ≤ c+2,, Glasnik Matematicki, 48, 291-299;, 2013
K. J. Batenburg, W. Fortes, L. HAJDU and R. Tijdemann,: Bounds on the quality of reconstructed images in binary tomography,, Disc. Appl. Math., 2013
A. Hajdu, L. HAJDU, L. Kovács, and H. Tomán,: Diversity measures for majority voting in the spatial domain,, HAIS, 2013
L. HAJDU, Á. PINTÉR, SZ. TENGELY and N. Varga,: Equal values of figurate numbers,, J. Number Theory, 2013
S. Brunetti, P. Dulio, L. HAJDU, C. Peri,: Discrete Tomography towards Denoising Ghosts,, közlésre benyújtva, 2013
A. BÉRCZES and V. Ziegler,: On geometric progressions on Pell equations and Lucas sequences,, Glasnik Matematicki, 48: 1-22,, 2013
A. BÉRCZES and A. PETHŐ,: On the sumset of Lucas sequences,, Publ. Math. Debrecen, közlésre benyújtva,, 2013
A. BÉRCZES and V. Ziegler,: On simultaneous palindromes,, Journal of Combinatorics and Number Theory, közlésre benyújtva,, 2013
G. NYUL and G. Rácz,: The r-Lah numbers,, közlésre benyújtva,, 2013
G. NYUL,: Pexiderized functional equations for vector products and quaternions,, Acta Math. Hungar., közlésre elfogadva,, 2013
N. Hirata-Kohno and T. KOVÁCS,: New algorithms to determine the S-integral points on elliptic curves,, Math. Comput., közlésre benyújtva,, 2013
A. Biró, K. GYŐRY, G. Harcos, J. Pintz, M. Simonovits and J. Szabados,: Number Theory, Analysis and Combinatorics, de Gruyter,, 2013
D. Dombek, L. HAJDU and A. PETHŐ,: Representing algebraic integers as linear combinations of units, Periodica Mathematica Hungarica, 68, 135-142,, 2014
S. Akiyama and A. PETHŐ,: On the distribution of polynomials with bounded roots II. Polynomials with integer coefficients, Unif. Distr. Theory, 9, 5-19,, 2014
S. Akiyama and A. PETHŐ,: On the distribution of polynomials with bounded roots I. Polynomials with real coefficients., J. Math. Soc. Japán, 66, 927-949,, 2014
I. GAÁL and G. Petrányi,: Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields,, Czech. Math. J., 64, 465-475,, 2014
L. HAJDU and V. Ziegler,: Distinct unit generated totally complex quartic fields,, Math. Comp. 83, 1495-1512,, 2014
L. HAJDU and M. Szikszai,: On the GCD-s of k consecutive terms of an elliptic divisibility sequence, Publ-Math. Debrecen, 84, 291-301,, 2014
A. BÉRCZES, A. Dujella and L. HAJDU,: Some diophantine properties of the sequence of S-units, J. Number Theory, 138, 48-68,, 2014
L. HAJDU and R. Tijdeman,: Bounds for approximate discrete tomography solutions, SIAM Journal on Discrete Mathematics, 27, 1055-1066,, 2013
L. Kovács, H. Tomán, Á. Jónás L. HAJDU and A. Hajdu,: Generalizing the majority voting scheme to spatially constrained voting,, IEEE Transactions on Image Processing, 22, 4182-4194,, 2013
G. NYUL, and Zs. Balogh,: Stirling numbers of the second kind and Bell numbers for graphs,, Australian J. of Combinatorics, 58, 264-274,, 2014
G. NYUL and E. Gyimesi,: A note on Golomb's method and the continued fraction method for Egyptian fractions, Annales Mathematicae et Informaticae, 42, 129-134,, 2013
A. BÉRCZES, J. H. Evertse and K. GYŐRY,: Effective results for Diophantine equations over finitely generated domains,, Acta Arith., 163, 71-100,, 2014
K. GYŐRY, T. KOVÁCS, Gy. Péter and Á. PINTÉR,: Equal values of standard counting polynomials,, Publ. Math. Debrecen, 84, 259-277, 2014
A. PETHŐ and SZ. TENGELY,: On composite rational functions, Turan Memorial Volume,, Number Theory, Analysis and Combinatorics de Gruyter, 241-260, De Gryter, Berlin, 2014
V. Komornik and A. PETHŐ,: Common expansions noninteger bases,, Publ. Math. Debrecen, 85, 489-501,, 2014
K. J. Batenburg, W. Fortes, L. HAJDU and R. Tijdemann,: Bounds on the quality of reconstructed images in binary tomography,, Disc. Appl. Math. 161, 2236-2251,, 2013
A. Hajdu, L. HAJDU, L. Kovács, and H. Tomán,: Diversity measures for majority voting in the spatial domain,, HAIS, Lecture Notes in Art. Int. 8073, 314-323, 2013
L. HAJDU, Á. PINTÉR, SZ. TENGELY and N. Varga,: Equal values of figurate numbers,, J. Number Theory, 137, 130-141,, 2014
A. BÉRCZES and A. PETHŐ,: On the sumset of binary recurrences sequences,, Publ. Math. Debrecen, 84, 279-290, 2014
A. BÉRCZES and V. Ziegler,: On simultaneous palindromes,, Journal of Combinatorics and Number Theory, 6,, 2014
G. NYUL and G. Rácz,: The r-Lah numbers,, Discrete Mathematics, 2013
G. NYUL,: Pexiderized functional equations for vector products and quaternions,, Acta Math. Hungar., 142, 519-525,, 2014
N. Hirata-Kohno and T. KOVÁCS,: S-integral points on elliptic curves via new approximation of p-adic elliptic logarithms, közlésre benyújtva, 2014
S. Akiyama, H. Brunotte, A. PETHŐ, W. Steiner and J. Thuswaldner,: Problems and conjectures around shift radix systems,, Open Problems in Mathematics, vol. 2, 2014
I. GAÁL and L. Remete,: Binomial Thue equations and power integral bases in pure quartic fields,, J P Journal of Algebra, Number Theory and Applications, 32, 49-61,, 2014
I. GAÁL, L. Remete and T. Szabó,: Calculating power integral bases by solving relative Thue equations,, Tatra. Mt. Math. Publ., 59, 1-13,, 2014
M. A. Bennett and Á. PINTÉR,: Intersections of recurrence sequences,, Proc. Amer. Math. Soc.,, 2014
Bo He, Á. PINTÉR, A. Togbé, N. Varga,: A generalization of a problem of Mordell, Glasnik Matematicki,, 2014
Bo He, Á. PINTÉR, A. Togbé,: On simultaneous Pell equation and related Thue equations, Proc. Amer. Math. Soc.,, 2014
L. HAJDU and I. Pink,: On the equation 1+2^a+x^b=y^n,, J. Number Theory, 143, 1-13,, 2014
K. GYŐRY, L. HAJDU, R. Tijdeman,: Representation of finite graphs as difference graphs of S-units,, I, J. Combin. Th. Series A, 127, 314-335,, 2014
A. BÉRCZES, L. HAJDU, N. Hirata-Kohno, T. KOVÁCS, A. PETHŐ,: A key exchange protocol based on Diophantine equations and S-integers,, JSIAM Letters, 2014
Cs. Bertók, L. HAJDU,: On a Hasse-type principle for exponential diophantine equations and its applications,, Math. Comp., 2014
L. HAJDU,: On a conjecture of Schaffer concerning the equation S_k(x)=y^n, közlésre benyújtva, 2014
L. HAJDU és M. Szikszai.,: Common factors in series of consecutive terms of associated Lucas and Lehmer sequences, közlésre benyújtva, 2014
A. Custic, L. HAJDU, D. Kreso, R. Tijdeman,: On conjectures and problems of Ruzsa, concerning difference graphs of S-units, közlésre benyújtva, 2014
S. Akiyama, L. Aszalós, L. HAJDU, A. PETHŐ,: Correlation clustering of graphs and integers, közlésre benyújtva, 2014
A. BÉRCZES: Effective results for unit points on curves over finitely generated domains,, Math. Proc. Cambridge Phil. Soc.,, 2014
A. BÉRCZES,: Effective results for division points on curves in G_m^2,, J. Théor. Nombres Bordeaux,, 2014
SZ. TENGELY, N. Varga,: On a generalization of a problem of Erdős and Graham,, Publ. Math. Debrecen, 84, 475-482,, 2014
A. BÉRCZES, A. Dujella, L. HAJDU, SZ. TENGELY,: Finiteness results for F-Diophantine sets,, közlésre benyújtva, 2014
N. Hirata-Kohno, T. KOVÁCS,: Computing S-integral points on elliptic curves of rank at least 3,, RIMS Kokyuroku 1898, Analytic Number Theory-Arithmetic Properties of Transcendental Functions and their Applications, 2013, RIMS, Kyoto University, Kyoto, 92-102,, 2014
N. Hirata-Kohno, T. KOVÁCS, T. Miyazaki,: On the Nagell-Ljunggren-eguation,, közlésre előkészítve, 2014
G. NYUL and B. Rauf,: On the existence of van der Waerden type numbers for linear recurrence sequences with constant coefficients,, Fibonacci Quarterly, közlésre elfogadva, 2014
E. Gyimesi and G. NYUL,: A note on combinatorial subspaces and r-Stirling numbers,, Utilitas Mathematica, közlésre elfogadva, 2014




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