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]**Ronald J. Evans,*Mathematical Notes: On Blocks of 𝑁 Consecutive Integers*, Amer. Math. Monthly**76**(1969), no. 1, 48–49. MR**1535230**, https://doi.org/10.2307/2316790**[2]**Jean-René Joly,*Nombre de solutions de certaines équations diagonales sur un corps fini*, C. R. Acad. Sci. Paris Sér. A-B**272**(1971), A1549–A1552 (French). MR**0282949****[3]**R. Kidl and H. Niederreiter,*Finite fields*, Encyclopedia of Math. and Its Appl., vol. 20, Addison-Wesley, Reading, Mass, 1983.**[4]**Wolfgang M. Schmidt,*Equations over finite fields. An elementary approach*, Lecture Notes in Mathematics, Vol. 536, Springer-Verlag, Berlin-New York, 1976. MR**0429733****[5]**Charles Small,*Diagonal equations over large finite fields*, Canad. J. Math.**36**(1984), no. 2, 249–262. MR**749983**, https://doi.org/10.4153/CJM-1984-016-6**[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