On The Diophantine Equation(47n)~x+(1104)~y ?(1105)~z

For odd modern first congruence equationax?b?modm?,we can get a complete set of residues prime to m through finitely multiplying 2.Therefore having i,such that2ⁱa₁?1?modm₁?,this equation must have solutions.Let a,b,c be relatively prime positive integers such thata ²+b²=c².In 1956,Jesmanowicz conjecture that for any positive integer n,the only solution of?a n?^x+?b n?^y=?c n?^z in positive integers is?x,y,z?=?2,2,2?.In this paper,we show that?47n?^x+?1104n?^y=?1105n?^z has no solution in positive integers other than?x,y,z?=?2,2,2?.In this paper,we show this conjecture for the case that a or b is 2^r1+1n₁ where r₁,n₁ is any positive integer.