On The Diophantine Equation (39n)~x + (760n)~y = (761n)~z

Let a,b,c be positive integers such tha any two among them are coprime and a2+b2=c2 In 1956,Jesmanowicz conjectured that the Diophantine equation(na)x+(nb)y=(nc)z has no solutions in positive integers other than x=y=z=2。By using elementary method,we proves that for any positive integer n the Diophantine equation(39n)x+(760n)y=(761n)z has no solutions in positive integers other than x=y=z=2。Jesmanowicz' conjecture is true when a=3.13,b=19.40,c=761。...