On the solvability of the equation and its application

Authors:
Qi Sun and Da Qing Wan

Journal:
Proc. Amer. Math. Soc. **100** (1987), 220-224

MSC:
Primary 11D04

DOI:
https://doi.org/10.1090/S0002-9939-1987-0884454-6

MathSciNet review:
884454

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Abstract: In this paper, we obtain a necessary and sufficient condition under which the equation of the title is unsolvable. More precisely, for the equation

1. For some , or

2. If is the set of all even integers among , then are pairwise prime, and is prime to any odd number in .

**[1]**R. J. Evans,*On blocks of**consecutive integers*, Amer. Math. Monthly**76**(1969), 48-49. MR**1535230****[2]**J. R. Joly,*Nombre de solutions de certaines équations diagonales sur un corps fini*, C. R. Acad. Sci. Paris Ser. A-B**272**(1971), 1549-1552. MR**0282949 (44:183)****[3]**R. Kidl and H. Niederreiter,*Finite fields*, Encyclopedia of Math. and Its Appl., vol. 20, Addison-Wesley, Reading, Mass, 1983.**[4]**W. M. Schmidt,*Equations over finite fields*, Lecture Notes in Math., vol. 536, Springer-Verlag, Berlin and New York, 1976. MR**0429733 (55:2744)****[5]**C. Small,*Diagonal equations over large finite fields*, Canad. J. Math.**36**(1984), 249-262. MR**749983 (86e:11094)****[6]**Sun Qi, Wan Daqing and Ma Degang,*On the equation*, Chinese Ann. of Math. (to appear).

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DOI:
https://doi.org/10.1090/S0002-9939-1987-0884454-6

Article copyright:
© Copyright 1987
American Mathematical Society