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2013•Applied Mechanics and MaterialsRequires access

Solutions for a Class of the Higher Diophantine Equation

Yin Xia Ran

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Abstract

We studied the Diophantine equation x2+4n =y7.By using the elementary method and algebaic number theroy, we obtain the following concusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 26n-1/7 contains some square factors. (ii) Let x be an even number, when n=7k(k≥1) , all integer solutions for the equation are (x,y)=(0,4k) ;when n=7k+3, all integer solutions are (x,y)=(±27k+3,22k+1); when n≡1,2,4,5,6 the equation has no integer solution.

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What this paper is about

We studied the Diophantine equation x2+4n =y7.By using the elementary method and algebaic number theroy, we obtain the following concusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 26n-1/7 contains some square factors. (ii) Let x be an even number, when n=7k(k≥1) , all integer solutions for the equation are (x,y)=(0,4k) ;when n=7k+3, all integer solutions are (x,y)=(±27k+3,22k+1); when n≡1,2,4,5,6 the equation has no integer solution.

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Available abstract

We studied the Diophantine equation x2+4n =y7.By using the elementary method and algebaic number theroy, we obtain the following concusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 26n-1/7 contains some square factors. (ii) Let x be an even number, when n=7k(k≥1) , all integer solutions for the equation are (x,y)=(0,4k) ;when n=7k+3, all integer solutions are (x,y)=(±27k+3,22k+1); when n≡1,2,4,5,6 the equation has no integer solution.

Key concepts: Diophantine equation, Integer (computer science), Mathematics, Diophantine set, Combinatorics, Class (philosophy), Discrete mathematics, Computer science

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