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A DIOPHANTINE PROBLEM CONCERNING POLYGONAL NUMBERS

Part of: Diophantine equations

Published online by Cambridge University Press:  25 January 2013

DAEYEOUL KIM ,
YOON KYUNG PARK  and
ÁKOS PINTÉR
DAEYEOUL KIM
Affiliation:
National Institute for Mathematical Sciences (NIMS), Daejeon 305-811, Korea email [email protected]
YOON KYUNG PARK
Affiliation:
School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, Korea email [email protected]
ÁKOS PINTÉR*
Affiliation:
Institute of Mathematics, MTA-DE Research Group ‘Equations, Functions and Curves’, Hungarian Academy of Sciences and University of Debrecen, P. O. Box 12, H-4010 Debrecen, Hungary
*
✉[email protected]
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Abstract

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Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

Keywords

Diophantine equationspolygonal numbers

MSC classification

Secondary: 11D41: Higher degree equations; Fermat's equation
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 88 , Issue 2 , October 2013 , pp. 345 - 350
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Bennett, M., ‘Products of consecutive integers’, Bull. Lond. Math. Soc. 36 (05) (2004), 683–694.CrossRefGoogle Scholar
Bennett, M. A., Győry, K., Mignotte, M. and Pintér, Á., ‘Binomial Thue equations and polynomial powers’, Compositio Math. 142 (05) (2006), 1103–1121.CrossRefGoogle Scholar
Bilu, Y. F., Brindza, B., Kirschenhofer, P., Pintér, Á. and Tichy, R. F., ‘Diophantine equations and Bernoulli polynomials’, Compositio Math. 131 (2) (2002), 173–188; with an appendix by A. Schinzel.CrossRefGoogle Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (3–4) (1997), 235–265; Computational algebra and number theory (London, 1993).CrossRefGoogle Scholar
Brindza, B., ‘On $S$-integral solutions of the equation ${y}^{m} = f(x)$’, Acta Math. Hungar. 44 (1–2) (1984), 133–139.CrossRefGoogle Scholar
Brindza, B. and Pintér, Á., ‘On equal values of power sums’, Acta Arith. 77 (1) (1996), 97–101.CrossRefGoogle Scholar
Brindza, B., Pintér, Á. and Turjányi, S., ‘On equal values of pyramidal and polygonal numbers’, Indag. Math. (N.S.) 9 (2) (1998), 183–185.CrossRefGoogle Scholar
Cassels, J. W. S., ‘Integral points on certain elliptic curves’, Proc. Lond. Math. Soc. (3) 14a (1965), 55–57.CrossRefGoogle Scholar
Deza, E. and Deza, M. M., Figurate Numbers (World Scientific, Hackensack, NJ, 2012).CrossRefGoogle Scholar
Dickson, L. E., History of the Theory of Numbers. Vol. II: Diophantine Analysis (Chelsea Publishing Co., New York, 1966).Google Scholar
Győry, K., Dujella, A. and Pintér, Á., ‘On the power values of pyramidal numbers, I’, Acta Arith. 155 (3) (2012), 217–226.Google Scholar
Győry, K. and Pintér, Á., ‘Binomial Thue equations, ternary equations and power values of polynomials’, J. Math. Sci. 180 (2012), 569–580.CrossRefGoogle Scholar
Kaneko, M. and Tachibana, K., ‘When is a polygonal pyramid number again polygonal?’, Rocky Mountain J. Math. 32 (1) (2002), 149–165.CrossRefGoogle Scholar
Krausz, T., ‘A note on equal values of polygonal numbers’, Publ. Math. Debrecen 54 (3–4) (1999), 321–325.CrossRefGoogle Scholar
Ljunggren, W., ‘Solution complète de quelques équations du sixième degré à deux indéterminées’, Arch. Math. Naturvid. 48 (7) (1946), 35.Google Scholar
Mihailescu, P., ‘Primary cyclotomic units and a proof of Catalan’s conjecture’, J. reine angew. Math. 572 (2004), 167–196.Google Scholar
Mordell, L. J., Diophantine Equations, Pure and Applied Mathematics, 30 (Academic Press, London, 1969).Google Scholar
Pintér, Á. and Varga, N., ‘Resolution of a nontrivial diophantine equation without reduction methods’, Publ. Math. Debrecen 79 (3–4) (2011), 605–610.CrossRefGoogle Scholar
Schinzel, A. and Tijdeman, R., ‘On the equation ${y}^{m} = P(x)$’, Acta Arith. 31 (2) (1976), 199–204.CrossRefGoogle Scholar