Solutions for a Class of the Exponential Diophantine Equation

Yin Xia Ran

doi:10.4028/www.scientific.net/AMR.753-755.3149

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HomeAdvanced Materials ResearchAdvanced Materials Research Vols. 753-755Solutions for a Class of the Exponential...

Solutions for a Class of the Exponential Diophantine Equation

Abstract:

We studied the Diophantine equation x²+4ⁿ=y¹¹. By using the elementary method and algebraic number theory, we obtain the following conclusions: (i) Let x be an odd number, one necessary condition which the equation has integer solutions is that 2^10n-1/11 contains some square factors. (ii) Let x be an even number, when n=11k(k≥1), all integer solutions for the equation are(x,y)=(0,4^k) ; whenn=11k+5(k≥0) , all integer solutions are(x,y)=(±2^11k+5,2^2k+1); when n≡1,2,3,4,6,7,8,9,10 the equation has no integer solution.