On the diophantine equation n(n+d)... (n+(k-1)d)=byl

被引:19
作者
Gyóry, K
Hajdu, L
Saradha, N
机构
[1] Hungarian Acad Sci, Number Theory Res Grp, H-4010 Debrecen, Hungary
[2] Univ Debrecen, Inst Math, H-4010 Debrecen, Hungary
[3] Tata Inst Fundamental Res, Sch Math, Bombay 400005, Maharashtra, India
来源
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES | 2004年 / 47卷 / 03期
关键词
D O I
10.4153/CMB-2004-037-1
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Oblath for the case of squares, and an extension of a theorem of Gyory on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in n, y when b = d = 1. We show that there are only finitely many solutions in n, d, b, y when k greater than or equal to 3, 1 greater than or equal to 2 are fixed and k + 1 > 6.
引用
收藏
页码:373 / 388
页数:16
相关论文
共 26 条
[1]   Ternary diophantine equations via galois representations and modular forms [J].
Bennett, MA ;
Skinner, CM .
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 2004, 56 (01) :23-54
[2]   ON THE EQUATIONS Z(M)=F(X,Y) AND AX(P)+BY(Q)=CZ(R) [J].
DARMON, H ;
GRANVILLE, A .
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 1995, 27 :513-543
[3]  
Darmon H, 1997, J REINE ANGEW MATH, V490, P81
[4]   PRODUCT OF CONSECUTIVE INTEGERS IS NEVER A POWER [J].
ERDOS, P ;
SELFRIDGE, JL .
ILLINOIS JOURNAL OF MATHEMATICS, 1975, 19 (02) :292-301
[5]  
Erdos P., 1939, J LOND MATH SOC, V14, P245
[6]  
Guy R. K., 1994, UNSOLVED PROBLEMS NU
[7]   NUMBER OF SOLUTIONS OF LINEAR EQUATIONS IN UNITS OF AN ALGEBRAIC NUMBER-FIELD [J].
GYORY, K .
COMMENTARII MATHEMATICI HELVETICI, 1979, 54 (04) :583-600
[8]  
Gyory K, 1998, ACTA ARITH, V83, P87
[9]  
GYORY K, 1999, POWER VALUES PRODUCT, P145
[10]   Effective majorations for the generalized Fermat equation [J].
Kraus, A .
CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 1997, 49 (06) :1139-1161