On The Diophantine Equation (21n)~x+(220n)~y=(221n)~z

Leta,b,c be relatively prime positive integers such that a2+b2=c2,Jesmanowicz conjectured in 1956 that for any given positive integer n the only solution of(an)x+ (bn)y=(cn)y in positive integers is x=y=z=2. In this paper,with the method of Legendre symbol,We have shown that for any positive integer n the Diophantine equation(21n)x+(220n)y=(221n)z has no solutions in positive in-teger other than x=y=z=2,that is, Jesmanowicz conjecture is true when a=2×10+1,b=2×10×(10+1),c=2×10×(10+1)=b+1...