Discussiones Mathematicae General Algebra and Applications 20(2) (2000) 193-198
doi: 10.7151/dmgaa.1016

[BIBTex] [PDF] [PS]


Zhenfu Cao

Department of Mathematics, Harbin Institute of Technology
Harbin 150001, P. R. China
e-mail: zfcao@hope.hit.edu.cn

Aleksander Grytczuk

Institute of Mathematics, Kotarbiński Pedagogical University
pl. Słowiański 6, 65-069 Zielona Góra, Poland
e-mail: agryt@lord.wsp.zgora.pl


In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.

Keywords: difference sets, diophantine equations, Pell's equations.

1991 Mathematics Subject Classification: 11D09, 05B10.


[1] K.T. Arasu, Recent results on difference sets, p. 1-23 in: ``Coding Theory and Design Theory'', Part II, Springer-Verlag, Berlin-New York 1990.
[2] Z. Cao, Some Diophantine equations in difference sets, a lecture in: ``5-th National Combinatorial Mathematics Conference", Shanghai 1994.
[3] Z. Cao, ``Introduction to Diophantine equations" (Chinese), Harbin Inst. of Technology Press, Harbin 1989.
[4] Z. Cao, On the equation axm-byn = 2, Chinese Sci. Bull. 35 (1990), 1227-1228.
[5] Z. Cao, On the Diophantine equation [(axm-4c)/(abx-4c)] = by2 (Chinese), J. Harbin Inst. Tech. 23 (1991), suppl., 110-112.
[6] G. Degert, Über die Bestimung der Grundeinheit gewisser reell-quadratischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 22 (1958), 92-97.
[7] Y.-D. Guo, On the exponential Diophantine equation x2 = 22ak2m-22akm+n+1, Discuss. Math.- Algebra & Stochastic Methods 16 (1996),57-60.
[8] S.L. Ma, McFarland's conjecture on abelian difference sets with multiplier -1, Des. Codes Cryptogr. 1 (1992), 321-332.
[9] L.J. Mordell, ``Diophantine Eqations", Academic Press, London 1969.
[10] C. Richaud, Sur la résolution des équations x2-Ay2 = ±, Atti Acad. Pontif. Nuovi Lincei, 1866, 177-182.
[11] C. Sto/rmer, Quelques theorems sur l'equation de Pell x2-Dy2 = ±1 et leur applications, Skr. Norske Vid. Acad., I Selsk. Mat. Natur. Kl., No. 2 (1897), 48 pp.

Received 11 March 1998
Revised 24 October 2000
Revised 4 December 2000