Diophantine Number Theory And its Applications  Page description

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Details of project

  
Identifier
75566
Type K
Principal investigator Pintér, Ákos
Title in Hungarian Diofantikus számelmélet és alkalmazásai
Title in English Diophantine Number Theory And its Applications
Keywords in Hungarian számelmélet, diofantoszi egyenletek, polinomok, kriptográfia, szomszédsági szekvenciák
Keywords in English number theory, diophantine equations, polynomials, cryptography, neighborhood sequences
Discipline
Mathematics (Council of Physical Sciences)100 %
Ortelius classification: Number theory
Panel Natural Sciences Committee Chairs
Department or equivalent Institute of Mathematics (University of Debrecen)
Participants Bazsó, András
Bérczes, Attila
Gaál, István
Győry, Kálmán
Hajdu, Lajos
Huszti, Andrea
Kovács, Tünde
Mező, István
Nyul, Gábor
Pink, István
Rakaczki, Csaba
Starting date 2009-01-01
Closing date 2012-12-31
Funding (in million HUF) 14.000
FTE (full time equivalent) 18.00
state closed project
Summary in Hungarian
Az utóbbi három évtizedben a debreceni számelméleti kutatócsoport fontos és nemzetközileg elismert szerepet játszott a diofantikus számelméletben. A pályázat keretében folytatjuk megkezdett kutatásainkat, és lehetőséget szeretnénk biztosítani a csoport fiatal generációjának. Az alapkutatásban nyert eredményeink következő alkalmazásaira koncentrálunk:

1) Aritmetikai gráfok és alkalmazásaik diofantikus problémákra.

2) A moduláris technika, Frey görbék és a Baker módszer alkalmazása klasszikus diofantoszi egyenletekre. Számítógépes eljárások fejlesztése az összes megoldás megkeresésére.

3) Effektív korlátok exponenciális egyenletek megoldásainak nagyságára, függvénytestek feletti diofantoszi egyenletek vizsgálata, index forma egyenletek, kombinatorikus számok aritmetikai tulajdonságai.

4) Alkalmazott Számelmélet: hash függvények és szomszédsági szekvenciák vizsgálata, kriptográfia.
Summary
In the last three decades the number theory research group in Debrecen played an important and internationally respected role in diophantine number theory. In the frame of this project we continue our previous research activity and intend to give a good opportunity for the young generation of the research group. We concentrate on the applications of our theoretical results:

1) Arithmetic graphs and applications to diophantine problems.

2) Applications of modular technique, Frey curves and Baker’s method to certain classical diophantine equations, development of computational techniques for solving them.

3) Effective bounds for the solutions of exponential diophantine equations, investigations of diophantine equations over function fields, index form equations and arithmetical properties of combinatorial numbers.

4) Applied Number Theory: investigations in the theory of hash functions, neighborhood sequences and cryptography.

In our investigations we apply almost every modern method of Diophantine Number Theory, combining them with strong computational techniques.




 

Final report

  
Results in Hungarian
A kutatócsoport tagjai elsősorban számelméleti, azon belül diofantikus egyenletekkel kapcsolatos vizsgálatokat folytattak. Több fontos egyenletosztály megoldására nyertek effektív, ineffektív és numerikus eredményeket, korábbi módszereket nem-triviális módon továbbfejlesztve és általánosítva. A kutatócsoport több tagja végzett alkalmazott számelméleti (diszkrét tomográfia és kriptográfia)i kutatásokat. Több eredmény született különböző klasszikus polinomok és polinomcsaládok gyökstruktúrájával és összetételével kapcsolatban. A pályázat ideje alatt, a pályázat támogatásával elért eredmények alapján 3 PhD (Bazsó, Huszti, Kovács) , 1 habilitáció (Bérczes) és 2 MTA Doktora (Hajdu, Pintér) fokozat született.
Results in English
The OTKA research group mainly dealt with numbertheoretical investigations. They gave several diophantine results including certain effective and ineffective finiteness statements and numerical results for the solutions to important families of diophantine equations extending and generalizing earlier methods. Some members studied numbertheoretical applications like the discrete tomography and cryptography. Other theorems related to the structure of zeros of polynomials and their decomposition are given. In the period 2009-2012, based on the results sponsored by this OTKA grant, the members obtained 6 new scientifc degree: 3 PhD degrees (Bazsó, Huszti, Kovács), 1 Habilitation (Bérczes) and 2 Doctor of MTA (Hajdu, Pinter).
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=75566
Decision
Yes





 

List of publications

  
A. Bazsó: On alternating power sums of arithmetic progressions, közlésre benyújtva, 2013
L. Hajdu, T. Kovács, A. Pethő, M. Pohst,: On an optimization problem for lattices,, Exper. Math. megjelenés alatt, 2013
A. Bérczes and I. Pink,: On the diophantine equation x^2+d^{21+1}=y^n,, Glasgow Math. Journal, 54, 415-428, 2012
L. Aszalós, A. Huszti,: Payment approval for PayWord, D.H. Lee, M. Yung (Eds.): 13th International Workshop on Information Security Applications 2012, Lecture Notes in Computer Science 7690, 161-176, Springe-Verlag, 2012
A. Huszti,: Experiment-based Definitions for Electronic Exam Systems,, Proceedings of the Mini-Conference on Applied Theoretical Computer Science, (2012) 127-130, 2010
L. Aszalós, A. Huszti,: Applying Spi-calculus for Payword,, Proceedings of ICAI'10 8th International Conference on Applied Informatics, (2012), 295-302, 2010
Bérczes A., Luca F.,: On the largest prime factor of numerators of Bernoulli numbers,, Indag. Math., 23, 128-134, 2012
Bérczes A., Luca F.,: On the sum of digits of numerators of Bernoulli numbers,, Canad. Math. Bull., közlésre elfogadva, 2013
Bérczes A., Dujella A., Hajdu L.: Some Diophantine properties of the sequence of S-units, közlésre benyújtva, 2013
G. Nyul and B. Rauf,: On the existence of van der Waerden type numbers for linear recurrence sequence with constant coefficients, benyújtás alatt, 2013
Zs. Kereskényi-Balogh and G. Nyul,: Stirling numbers of the second kind and Bell numbers for graphs,, közlésre benyújtva, 2013
E. Gyimesi and G. Nyul,: A note on Golomb's method and the continued fraction method for Egyptian fractions,, közlésre benyújtva, 2013
B. Nyul and G. Nyul,: Functional equations for vector products and quaternions,, Aequationes Mathematicae, közlésre elfogadva, 2013
Győry K. and Pintér Á.: Binomial Thue equations, ternary equations and power values of polynomials,, Fundam. Prikl. Math. 16, 61-77, 2010
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., 2012
K. Győry and Á. Pintér,: Binomial Thue equations, ternary equations and power values of polynomials,, J. Math. Sciences 180, 569-580,, 2012
A Bérczes, J.H. Evertse and K. Győry,: Multiply monogenic orders,, Ann Scuola Normale Sup. Pisa, közlésre elfogadva, 2013
A. Bérczes, J.H. Evertse and K. Győry,: Effective results for hyper- and superelliptic equations over number fields,, Publ. math. Debrecen, 2013
K. Győry,: Perfect powers in products with consecutive terms arithmetic progressions, II,, megjelenés alatt, 2013
A. Bérczes, J.H. Evertse and K. Győry,: Effective results for diophantine equations over finitely generated domains,, Acta Arith., megjelenés alatt, 2013
I. Gaál and M. Pohst,: Diophantine equations over global function fields IV: S-unit equations in several variables with an applications to norm form equations, J. Number Theory 130, 493-506,, 2010
I. Gaál and T. Szabó,: A note on the minimal indices of pure cubic fields, JP Journal of Algebra, Number Theory and Applications, 19, 129-139,, 2010
I. Gaál and G. Petrányi: Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields, benyújtott, 2013
I. Gaál and T. Szabó: Relative power integral bases in infinite families of quartic extensions quadratic field, benyújtott, 2013
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 Appell polynomials and the associated classes of polynomial expansions,, Aequationes Mathematicae, DOI 10.1007/s00010-012-0148-8, 2013
Y. Bilu, C. Fuchs, F. Luca and Á. Pintér: Combinatorial diophantine eguations 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, to apper in, Bulletin of Australian Math. Society, 2013
Á. Pintér and Sz. Tengely: The Korteweg-de Vries equation and a diophantine problem, benyújtott, 2013
Mező I. and Dil A.,: Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibinacci sequence, Cent. Eur. J. Math. 7(2), 310-321,, 2009
Fenyvesi A., Valastyan I., Mező I., Fenyvesi E.: Molecular dynamics simulation of radiation damage in beryllium, Hungarian Academy of Sciences, Budapest p.30, 2009
Fenyvesi A., Valastyan I., Mező I., Fenyvesi E,: Molecular dynamics simulation of radiation damage in beryllium, Hungarian Academy of Sciences, Budapest p.26, 2009
Mező I.: A q-Raabe formula and an integral of the fourth Jacobi theta function, J. Number Th. 133 (2), 692-702, 2012
Mező I.: The Primorial and the Riemann zeta function,, Amer. Math. Monthly, közlésre elfogadva, 2013
Mező I.,: Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions, Cent Eur. J. Math., közlésre elfogadva, 2013
Mező I.,: Some infinite sums arising from the Weierstrass Product Theorem, Applied. Math. Comput. közlésre elfogadva, 2013
Hajdu L. és R. Tijdeman,: Representing integers as linear combinations of power products,, Arch. Math. 98, 527-533, 2012
I. Mező: The r-Bell numbers,, J. Integer Seq. 14(1) Article 11.1.1, 2011
I Mező: The Dual of Spivey's Bell Number Formula, J. Integer Seq. 15(2) Article 12.2.4, 2012
I. Mező and K. Nagy: Integrals of weighted maximal logarithmic kernels on bounded Vilenkin groups, , Period. Math. Hungar., 64, 89-99, 2012
I. Mező: Duplication formulae involving Jacobi theta functions and Gosper's q-trigonometric functions,, Proc. AMS, megjelenés alatt, 2012
Á. Pintér and N. Varga: Resolution of a nontrivial diophantine equation without reduction methods, Publ. Math. Debrecen 79:(3-4) pp. 605-610., 2011
J Ferenczik, Á Pintér, B Porvázsnyik: On Equal Values of Stirling Numbers of the Second Kind., APPLIED MATHEMATICS AND COMPUTATION 218:(3) pp. 980-984., 2011
M. A. Bennett, I. Pink and Zs. Rábai: On the number of solutions of some binomial Thue equations, Publ. Math. Debrecen, közlésre benyújtva, 2012
L. Hajdu and R. Tijdeman: Representing integers as linear combinations of powers, Publ. Math. Debrecen 79, 461--468., 2011
A. Bérczes, A. Dujella, L. Hajdu and F. Luca: On the size of sets whose elements have perfect power n-shifted products, Publ. Math. Debrecen 79, 325-339., 2011
Kovács L., Tomán H., Jónás Á., Hajdu L. and Hajdu A: Generalizing the majority voting scheme to conditional voting, Lecture Notes in Artificial Intelligence 6679, 189-196., 2011
L. Hajdu, N. Saradha and R. Tijdeman: On a conjecture of Pomerance, Acta Arithmetica, 155, 175-184, 2012
B. van Dalen, L. Hajdu and R. Tijdeman: Bounds for discrete tomography solutions, közlésre benyújtva, 2012
L. Hajdu and M. Szikszai: On the GCD-s of k consecutive terms of Lucas sequences, J. Number Theory 132, 3056-3069, 2012
D. Dombek, Hajdu L. and Pethő A: Representing algebraic integers as linear combinations of units, közlésre benyújtva, 2012
J. Batenburg, W. Fortes, Hajdu L. and R. Tijdeman: Bounds on the difference between reconstructions in binary tomography, Discrete Geometry and ComputerImaginary LNCS 6607, 369-380, 2011
A. Huszti, Z. Kovács: Univerzális kriptográfiai protokoll e-felmérésekhez, Informatika a Felsőoktatásban Konferencia 2011, 592--600., 2011
J. Folláth, A. Huszti, A. Pethő: Informatikai biztonság és kriptográfia, Egytemi jegyzet, DE, 2011
Tomán H., Kovács L., Jónás Á., Hajdu L. és Hajdu A.,: Generalized weighted majority voting with an application to algorithms having spatial output, 7th Int. Conf. on Hybrid Artificial Intelli-gence Systems (HAIS 2012),, Lecture Notes in Computer Science, 7209 pp. 56-67,, 2012
Á. Pintér: On a class of diophantine equations related to the numbers of cells in hyperplane arrangements, Journal of Number Theory 129: pp. 1664-1668. (2009), 2009
Liptay K, Luca F, Pinter A, Szalay L: Generalized balancing numbers, INDAGATIONES MATHEMATICAE-NEW SERIES 20: pp. 87-100. (2009), 2009
Cangul IN, Demirci M, Luca F, Pintér Á, Soydan G: On the diophantine equation x^2+2^a 11^b=y^n, FIBONACCI QUARTERLY 48: pp. 39-46. (2010), 2010
Gy Péter, Á Pintér, A Schinzel: On equal values of trinomials, MONATSHEFTE FÜR MATHEMATIK 162: pp. 313-320., 2011
A. Bazsó: Thue equations and ternary equations with S-unit coefficients, Publ.Math.Debrecen, 77, 499-516, 2010
A.Bazsó: Binomial Thue equations, ternary equations and their applications, PhD értekezés, Debrecen, 2010
A Bérczes, J-H. Evertse and K. Győry: On monogenic orders over domains finitely generated over Z, közlére előkészítve, 2011
A. Bérczes and V. Ziegler: On geometric progressions in the solution set of Pell equations and in Lucas sequences, közlésre előkészítve, 2011
T. Kovács, K. Liptai and P. Olajos: On (a,b) balancing numbers, Publ. Math. Debrecen 77, 485-498, 2010
T. Kovács, Gy. Péter, N. Varga: On some polynomial values of repdigit numbers, Periodica. Math. Hungar., megjelenés alatt, 2012
Cs. Rakaczki: On some generalization of diophantine equation s(1^k+...+x^k)+r=dy^n, Acta Arithmetica, 151, 201-216, 2012
K.J.Batenburg, W.Fortess, L. Hajdu and R. Tijdeman: Bounds on the difference between reconstruction in binary tomography, Discrete Geometry and Computer Imaginary, LNCS 6607, 369--380., 2011
L. Hajdu and N. Saradha: Disproof of a conjecture of Jacobsthal, Math. Comp. 81, 2461-2471, 2012
A. Hajdu, L. Hajdu and R. Tijdeman: Approximation of Euclidean distances by chamfer distances, Acta Cybernetica 20, 399-417, 2012
I. Gaál and M. Pohst: Solving explicitly diophantine equations of type $F(x,y)=G(x,y)$ over function fields, Funct. Approx. Comment. Math. 45, 79--88., 2011
I. Gaál and M. Pohst: The sum of two S-units being a perfect power in global function fields, Math. Slovaka, megjelenés alatt, 2011
K. Győry: S-unit equations in number fields: effective results, generalizations, abc-conjecture, Analytic number theory and related topics, Kyoto, RIMS, No. 1710 (2010), 71-84., 2010
K.Győry and C. Smyth: The divisibility of a^n - b^n by powers of n, Integers 10, 319-334., 2010
K.Győry, L.Hajdu and R.Tijdeman: Irreducibility criteria of Schur type and Pólya-type, Monatshefte Math., Monatsh. Math. 163, 415-443., 2011
I. Mező: A new formula for the Bernoulli polynomials, Results Math. 58, 329-355, 2010
A. Pintér and Alf van der Poorten: A simple observation on simple zeros, Archiv der Math. 95, 355-361, 2010
Grenczer M, Pinter A, Zsuga J, Kemeny-Beke A, Juhasz B, Szodoray P, Tosaki A, Gesztelyi R: The influence of affinity, efficacy, and slope factor on the estimates obtained by the receptorial responsiveness method (RRM): A computer simulation study., CANADIAN JOURNAL OF PHYSIOLOGY AND PHARMACOLOGY 88: pp. 1061-1073, 2010
Grenczer M, Zsuga J, Majoros L, Pinter A, Kemeny-Beke A, Juhasz B, Tosaki A, Gesztelyi R: Effect of asymmetry of concentration-response curves on the results obtained by the receptorial responsiveness method (RRM): An in silico study., CANADIAN JOURNAL OF PHYSIOLOGY AND PHARMACOLOGY 88: pp. 1074-1083, 2010
I. Pink and Zs. Rábai,: On the diophantine equation $x^2+5^k17^l=y^n$, Communications in Mathematics, 19, 1-9,, 2011
A. Bérczes and I. Pink: On the diophantine equation $x^2+d^{2l+1}=y^n$, Glasgow Math. Journal, közlésre benyújtva, 2011
A. Bazsó, Á. Pintér, H. M. Srivastava: A refinement of Faulhaber’s theorem concerning sums of powers of natural numbers,, Applied Math. Letters 25:(3) pp. 486-489., 2012
A. Bazsó, D. Kreso, F. Luca, Á. Pintér: On equal values of power sums of arithmetic progressions, Glasnik Matematicki, 47, 253-263, 2012
L. Hajdu, T. Kovács: Almost fifth powers in arithmetic progression, J. Number Theory 131, 1912-1923., 2011
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
Claus Fieker, István Gaál and Michael Pohst,: On computing integral points of a Mordell curve over rational function fields in characteristic $>3$, J. Number Theory 133, 738-750., 2013
Andrea Huszti: A Homomorphic Encryption-Based Secure Electronic Voting Scheme, Publicationes Mathematicae Debrecen, 79/3-4, 479 -- 496., 2011
Andrea Huszti, Attila Pethő: A Secure Electronic Exam System, Publicationes Mathematicae Debrecen, 77/3-4, 299-312, 2010
Zs. Ádám, L. Hajdu and F. Luca: Representing integers as linear combinations of S-units, Acta Arith. 138 (2009), 101-107., 2009
L. Hajdu and T. Kovács: Parallel LLL-reduction for bounding the integral solutions of elliptic equations,, Math. Comp. 78 (2009), 1201-1210., 2009
L. Hajdu, Sz. Tengely and R. Tijdeman: Cubes in products of terms in arithmetic progression, Publ. Math. Debrecen 74 (2009), 215-232., 2009
K. Győry, L. Hajdu and Á. Pintér: Perfect powers from products of consecutive terms in arithmetic progression, Compositio Math. 145 (2009), 845-864., 2009
L. Hajdu: Optimal systems of fundamental S-units for LLL-reduction, Period. Math. Hungar. 59 (2009), 79-105., 2009
L. Hajdu, A. Schinzel and M. Skalba: Multiplicative properties of sets of positive integers, Archiv der Math. 93 (2009), 269-276., 2009
L. Hajdu and Sz. Tengely: Arithmetic progressions of squares, cubes and n-th powers, J. Functiones et Approximatio 41 (2009), 129-138., 2009
A. Bérczes, L. Hajdu L. and A. Pethő: Arithmetic progressions in the solution sets of norm form equations, Rocky Mountain J. Math. 40, 383-395, 2010
L. Hajdu and F. Luca: On the length of arithmetic progressions in linear combinations of S-units, Archiv der Math., 94, 357-363, 2010
L. Hajdu and N. Saradha: On a problem of Pillai and its generalizations, Acta Arith. 144, 323-347., 2010
L. Hajdu and N. Saradha: On a problem of Recaman and its generalization, J. Number Theory 131, 18--24., 2011
Hajdu L.: Számtani sorozatok multiplikatív tulajdonságú halmazokban, MTA doktori értekezés, benyújtva, 2010
I.Gaál and M.Pohst: Diophantine equations over global function fields V:Resultant equations in two unknown polynomials, Int. J. Pure Appl. Math. 53 (2009), 307-317., 2009
I.Gaál and M.Pohst: On solving norm equations in global function fields, J. Math. Cryptology 237-248, 2009
T. Kovács: Combinatorial numbers in binary recurrences, Periodica Mathematica Hungarica, 58 (2009) 83-98., 2009
G. Nyul: Field indices of some well-known family of algebraic number fields, közlésre előkészítve, 2010
G. Nyul and B. Rauf: Upper bounds on van der Waerden type numbers for some second order linear recurrence, Annales Mathematicae et Informaticae 38, 117-122., 2011
I. Mező: Analytic extension of hyperharmonic numbers, Online J. Anal. Comb 4 (2009), 2009
I. Mező: Several generating functions for second-order recurrence sequences, J. Integer Seq. 12(3) (2009) Article 09.3.7., 2009
A. Dil and I. Mező: Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130(2) (2010), 360-369., 2010
M. Griffiths and I. Mező: A generalization of Stirling numbers of the second kind via multisets, J. Integer Seq.13(2) 2010, Article 10.2.5., 2010
Cs. Rakaczki: On some diophantine results related to Hermite polynomials, Functiones et Approcimatio Comment. Math. 42, 7-16., 2010
Cs. Rakaczki: On the simple zeros of shifted Euler polynomials, Publ. Math. Debrecen, 79/3-4, 623-636, 2011
Bérczes A, Evertse J-H, Győry K: Effective results for linear equations in two unknowns from a multiplicative division group, Acta Arith., 2009; 136: 331-349, 2009
Bérczes A, Evertse J-H, Győry K, C. Pontreau: Effective results for points on certain subvarieties of tori, Math. Proc. Cambridge Phil. Soc., 2009, 147: 69-94., 2009
Bérczes A, Járási I.: On the application of index forms in cryptography, Periodica Math. Hungar., 2009; 58:35–45., 2009
Bérczes A.: On the sumsets of geometric progressions, Publ. Math. Debrecen 77, 261-276., 2010
Bazsó A, Bérczes A, Győry K, Pintér Á.: On the resolution of equations Ax^n-By^n=C in integers x, y, and n≥3,II, Publ. Math. Debrecen 76, 227-250, 2010
Bérczes A, Folláth J, Pethő: On a family of collision-free functions, Tatra Mt. Math. Publ., 47, 1-13, 2010
Bérczes A., Liptai K., Pink I.: On balancing recurrence sequences, Fibonacci Quarterly 48, 121-128, 2010



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