Andrej Dujella:

Diofantske jednadžbe - Seminarske teme


1. Kompaktna reprezentacija i algoritmi za rješavanje Pellovih jednadžbi (Filip Najman, 13.6.2007.)

H. W. Lenstra: Solving the Pell Equation, Notices Amer. Math. Soc. 49 (2002), 182-192.

H. Williams: Solving the Pell equation, in: Number theory for the millennium, III (Urbana, IL, 2000), A K Peters, Natick, 2002, pp. 397-435.

M. J. Jacobson Jr., H. C. Williams: Modular arithmetic on elements of small norm in quadratic fields, Des. Codes and Cryptogr. 27 (2002), 93-110.

H. Cohen: A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993. (Poglavlja 5.8, 5.9)


2. Kriteriji za rješivost pellovskih jednadžbi x2 - dy2 = ± 2 (Ivan Soldo, 20.6.2007.)

R. A. Mollin: Lagrange, central norms, and quadratic Diophantine equations, International J. Math. and Math. Sci. 7 (2005), 1039–1047.

H. Yokoi: Solvability of Diophnatine equations x2 - dy2 = ± 2 and new invariants for real quadratic fields, Nagoya Math. J. 134 (1994), 137-149.

F. Lemmermeyer: Higher descent on Pell conics. I. From Legendre to Selmer, preprint, math.NT/0311309.


3. Polinomijalna Pellova jednadžba (Vinko Petričević, 11.7.2007.)

M. B. Nathanson: Polynomial Pell’s equations, Proc. Amer. Math. Soc. 86 (1976), 89–92.

J. Steuding: Diophantine Analysis, Chapman & Hall/CRC, Boca Raton, 2005. (Poglavlja 6.7, 6.8)

W. A. Webb, H. Yokota: Polynomial Pell’s equation, Proc. Amer. Math. Soc. 131 (2003), 993–1006.

W. A. Webb and H. Yokota: Polynomial Pell’s equation–II, J. Number Theory 106 (2004), 128–141.

J. Mc Laughlin: Polynomial solutions to Pell's equation and fundamental units in real quadratic fields, J. London Math. Soc. (2) 67 (2003), 16-28.

A. M. S. Ramasamy: Polynomial solutions for the Pell’s equation, Indian J. Pure Appl. Math. 25 (1994), 577–581.


4. Sume tri kvadrata (Goran Conar, 2.5.2007.)

J.-P. Serre: A Course in Arithmetic, Springer-Verlag, New York, 1996. (Poglavlje IV-Appendix)

E. Landau: Elementary Number Theory, Chelsea, New York, 1966. (Part III - Poglavlje IV)

W. Sierpinski: Elementary Theory of Numbers, PNW, Warszawa; North Holland, Amsterdam, 1987. (Poglavlje IX.4)


5. Kontraprimjeri za lokalno-globalni princip (Marcel Maretić, 9.10.2013.)

J. W. S. Cassels: Lectures on Elliptic Curves, Cambridge University Press, Cambridge, 1995. (Poglavlje 18)

H. Cohen: Explicit methods for solving Diophantine equations, Lecture notes, Arizona Winter School 2006.


6. Rezultante polinoma i sustavi polinomijalnih jednadžbi (Tomislav Pejković, 11.7.2007.)

D. Cox, J. Little, D. O’Shea: Using Algebraic Geometry, Springer, New York, 2005. (Poglavlje 3)

A. Schinzel: Selected Topics on Polynomials, University of Michigan Press, Ann Arbor, 1982. (Poglavlje 9)

W. M. Schmidt: Equations over Finite Fields: An Elementary Approach, Springer-Verlag, Berlin, 1976. (Poglavlje V.1)

A. Dujella and F. Luca: Diophantine m-tuples for primes, Intern. Math. Research Notices 47 (2005), 2913-2940.


7. Visine algebarskih brojeva (Luka Lasić, 3.10.2007.)

N.-P. Skoruppa: Heights, Lecture Notes, Universität Siegen

Y. Bugeaud: Approximation by Algebraic Numbers, Cambridge University Press, 2004. (Poglavlje 11)

M. Waldschmidt: Introduction to recent results in transcendental number theory, Workshop and conference in number theory, Hong-Kong, 1993.

C. Stewart: Linear Forms in Logarithms and Diophantine Equations, Lecture Notes, University of Waterloo.


8. Sedmi Hilbertov problem (Ana Ban, 24.11.2010.)

W. J. LeVeque: Topics in Number Theory. Volume II, Dover, New York, 2002. (Poglavlja 5-6, 5-7)

A. Baker: Transcendental Number Theory, Cambridge University Press, 1990. (Poglavlje 2)

H. E. Rose: A Course in Number Theory, Oxford University Press, 1995. (Poglavlje 8.3)


9. Padéovi aproksimanti i simultane diofantske aproksimacije

J. H. Rickert: Simultaneous rational approximations and related diophantine equations, Math. Proc. Cambridge Philos. Soc. 113 (1993), 461–472.

M. A. Bennett: Simultaneous rational approximation to binomial functions, Trans. Amer. Math. Soc. 348 (1996), 1717-1738.

M. A. Bennett: On the number of solutions of simultaneous Pell equations, J. Reine Angew. Math. 498 (1998), 173-200.

B. Jadrijević and V. Ziegler: A system of relative Pellian equations and a related family of relative Thue equations, Int. J. Number Theory 2 (2006), 569-590.


10. Varijante Baker-Davenportove redukcije

A. Baker and H. Davenport: The equations 3x2 - 2 = y2 and 8x2 - 7 = z2, Quart. J. Math. Oxford Ser. (2) 20(1969), 129–137.

B. M. M. de Weger: Algorithms for Diophantine Equations, Centrum voor Wiskunde en Informatica, Amsterdam, 1989. (Poglavlje 3)

A. Pethö: Algebraische Algorithmen, Vieweg, Braunschweig, 1999. (Poglavlje 5.9)

I. Gaal: Diophantine Equations and Power Integral Bases, Birkhäuser, Boston, 2002. (Poglavlje 2.2)

A. Dujella and A. Pethö: A generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49 (1998), 291-306.

A. Dujella: A proof of the Hoggatt-Bergum conjecture, Proc. Amer. Math. Soc. 127 (1999), 1999-2005.


11. Grinsteadova metoda za rješavanje simultanih pellovskih jednadžbi

C. M. Grinstead: On a method of solving a class of Diophantine equations, Math. Comp. 32 (1978), 936-940.

R. G. E. Pinch: Simultaneous Pellian equations, Math. Proc. Camb. Phil. Soc. 103 (1988), 35-46.

E. Brown: Sets in which xy + k is always a square, Math. Comp. 45 (1985), 613-620.

N. Tzanakis: Explicit solution of a class of quartic Thue equations, Acta Arith. 64 (1993), 271–283.

A. Dujella: The problem of the extension of a parametric family of Diophantine triples, Publ. Math. Debrecen, 51 (1997), 311-322.


12. Ramanujan-Nagellova jednadžba (Kristina Krulić, 16.5.2007.)

W. Johnson: The Diophantine equation X2 + 7 = 2n, Amer. Math. Monthly 94 (1987), 59-62.

I. Stewart, D. Tall: Algebraic Number Theory, A K Peters, Natick, 2002. (Poglavlje 4.9)

L. J. Mordell: Diophantine Equations, Academic Press, London, 1969. (Poglavlje 23)

Y. Bugeaud, M. Mignotte, S. Siksek: Classical and modular approaches to exponential Diophantine equations. II. The Lebesgue-Nagell Equation, Compositio Math., to appear.


13. Delaunay-Nagellov teorem o jednadžbi x3 - dy3 = 1 (Danijel Krizmanić, 5.9.2007.)

W. J. LeVeque: Topics in Number Theory. Volume II, Dover, New York, 2002. (Poglavlja 3-7, 3-8, 3-9)

L. J. Mordell: Diophantine Equations, Academic Press, London, 1969. (Poglavlja 23, 24)

E. J. Barbeau: Pell's Equation, Springer-Verlag, New York, 2003. (Poglavlje 7)

S. Akiyama, H. Brunotte and A. Pethö: Cubic CNS Polynomials, notes on a conjecture of W.J.Gilbert, J. Math. Anal. and Appl., 281 (2003), 402-415.


14. Hardy-Ramanujanov problem taksija (Romana Capor)

J. H. Silverman, J. Tate: Rational Points on Elliptic Curves, Springer-Verlag, New York, 1992. (Poglavlje V.2)

J. H. Silverman: Taxicabs and sums of two cubes, Amer. Math. Monthly 100 (1993), 331-340.

D. W. Wilson: The fifth taxicab number is 48988659276962496, Journal of Integer Sequences 2 (1999), Article 99.1.9.

P. Emelyanov: On hunting for taxicab numbers, preprint, 2008.


15. Nepostojanje točaka reda 11 na eliptičkim krivuljama nad Q (Andrijana Radovčić, 5.9.2007.)

I. Kiming: There are no points of order 11 on elliptic curves over Q, Lecture notes, University of Copenhagen, 2006.

G. Everest, T. Ward: An Introduction to Number Theory, Springer-Verlag, London, 2005. (Poglavlje 5.4)


16. Veliki Fermatov teorem za parne eksponente (Marina Ninčević, 16.9.2009.)

P. Ribenboim: 13 Lectures on Fermat's Last Theorem, Springer-Verlag, New York, 1979. (Poglavlje IV.6)

P. Ribenboim: Fermat's Last Theorem for Amateurs, Springer-Verlag, New York, 2000. (Poglavlje VI)


17. Diofantske jednadžbe xp + yq = zr (Biserka Draščić Ban, 10.10.2007.)

H. Cohen: Number Theory. Volume II: Analytic and Modern Tools, Springer Verlag, Berlin, 2007. (Poglavlje 14)

H. Cohen: Explicit methods for solving Diophantine equations, Lecture notes, Arizona Winter School 2006.

N. Bruin: Chabauty methods and covering techniques applied to generalised Fermat equations, PhD-thesis, University of Leiden, 1999.


18. Specijalni slučajevi Catalanove slutnje (Neda Lovričević, 21.11.2007.)

H. Cohen: Number Theory. Volume I: Tools and Diophantine Equations, Springer Verlag, Berlin, 2007. (Poglavlje 6.11)

R. Schoof: Catalan's Conjecture, Lecture Notes, Universita di Roma "Tor Vergata".

L. J. Mordell: Diophantine Equations, Academic Press, London, 1969. (Poglavlje 30)

J. Daems: A cyclotomic proof of Catalan's conjecture, Master Thesis, Leiden University, 2003.


19. Rungeova metoda za diofantske jednadžbe (Ljerka Jukić, 12.12.2007.)

F. Beukers, Sz. Tengely: An implementation of Runge's method for Diophantine equations, preprint, math.NT/0512418.

Sz. Tengely: Effective Methods for Diophantine Equations, PhD thesis, Leiden University, 2004.

P. G. Walsh: A quantitative version of Runge’s theorem on Diophantine equations, Acta Arith. 62 (1992), 157-172. (errata)


20. Konačnost broja rješenja diofantske jednadžbe f(x) = g(y) (Sintija Tropper)

Y. Bilu and R. F. Tichy: The Diophantine equation f(x) = g(y), Acta Arith. 95 (2000), 261-288.

A. Schinzel: Selected Topics on Polynomials, University of Michigan Press, Ann Arbor, 1982. (Poglavlja 5, 6)

Th. Stoll: Finiteness Results for Diophantine Equations Involving Polynomial Families, PhD Dissertation, Technische Universität Graz, 2003.

A. Dujella and I. Gusić: Indecomposability of polynomials and related Diophantine equations, Q. J. Math. (Oxford) 57 (2006), 193-201.

Th. Stoll: Complete decomposition of Dickson-type polynomials and related Diophantine equations, preprint.

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