The Study On Je(?)manowicz’ Conjecture

In 1956,L.Jesmanowicz conjectured that,for any positive integer n,the exponential Diophantine equation(an)x+(bn)y-(cn)2 has only the positive integer solution(2,2,2),where a,b,c are positive integers satisfying a2+b2=c2 and gcd(a,b,c)=1.In this paper,using methods of elementary number theory and algebra number theory,we obtain the following results on the basics of works by Miyazaki et al:(1)Let a,b,c are positive integers satisfying a2+b2=c2 and gcd(a,b,c)=1.If a≡-1(mod b)and n is a positive integer with P(a)|n,where P(a)denotes the product of distinct prime factors of a,then Jesmanowicz’ conjecture is true.In particular,if(a,b,c)=(119,120,169),then Jesmanowicz’ conjecture is true.(2)Let(a,b,c)=(p2-16,8p,p2+16),where(?)is a prime number greater than three,with(?)(mod 12).If n is a positive integer with P(n)| b,then Jesmanowicz’conjecture is true.(3)Let(a,b,c)=(p2-16,8p,p2+16),where p is a prime with p≡11(mod 12).If n=1,then Jesmanowicz’ conjecture is true.(4)Let(a,b,c)=(2g+1,2g(g+1),2g2+29+1),r=6k+2,where k∈N,and k≥25.If n≡0,6,9(mod 12),and g=2r-1,then Jesmanowicz’ conjecture is true.