Affiliation:
1. College of Mathematics and Statistics, Yulin University, Yulin, Shaanxi 719000, China
2. School of Mathematics, Northwest University, Xi’an, Shaanxi 710127, China
Abstract
Letpbe a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) ifp≡±3(mod 8), then the equation8x+py=z2has no positive integer solutions(x,y,z); (ii) ifp≡7(mod 8), then the equation has only the solutions(p,x,y,z)=(2q-1,(1/3)(q+2),2,2q+1), whereqis an odd prime withq≡1(mod 3); (iii) ifp≡1(mod 8)andp≠17, then the equation has at most two positive integer solutions(x,y,z).
Funder
National Natural Science Foundation of China
Subject
General Environmental Science,General Biochemistry, Genetics and Molecular Biology,General Medicine
Cited by
2 articles.
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1. On the Diophantine Equation 8^x+n^y=z^2;WSEAS TRANSACTIONS ON MATHEMATICS;2020-10-30
2. On the Diophantine Equation (4^n)^x − P^y = Z^2;WSEAS TRANSACTIONS ON MATHEMATICS;2020-06-16