Solutions for a Class of the Higher Diophantine Equation

Yin Xia Ran

doi:10.4028/www.scientific.net/AMM.336-338.2265

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HomeApplied Mechanics and MaterialsApplied Mechanics and Materials Vols. 336-338Solutions for a Class of the Higher Diophantine...

Solutions for a Class of the Higher Diophantine Equation

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)=(0,4^k) ;when n=7k+3, all integer solutions are (x,y)=(±2^7k+3,2^2k+1); when n≡1,2,4,5,6 the equation has no integer solution.