The Diophantine Equation (51n)~x+(1300n)~y=(1301n)~z

Indefinite equation have high position in number theory,however,solving the index of Indefinite equation ax + by = cz is difficult.In 1956,Je(?)manowicz con-jectured that the Diophantine equation(an)x +(bn)y =(cn)z has only the integer solution(x,y,z)=(2,2,2),where a,b,c are positive integers that are pairwise prime and satisfy a2 + b2 = c2.In this paper,using the elementary method shows that:for any positive integer n,indefinite equation(51n)x +(1300n)u =(1301n)z has only the positive solution(x,y,z)=(2,2,2).that is the conjecture of Je(?)manowicz holds when a = 51,b = 1300,c = 1301.