About The Je(?)manowicz Conjecture

In 1956, Jesmanowicz posed the following conjecture:for any positive inte-ger n, the equation as only the positive integer solution where a, b, care positive integers satisfying andIn 1959, Lu proved that the equation has only the positive integer solution( In 2014, Deng proved that, for and any positive integer n, if P(a)|n or P(n)|a, where P(n) denotes the product of distinct prime factors of n, then the Jesmanowicz'conjecture is true.In this paper, we obtain the following results:(1) ,where k is a positive integer. If n is a positive integer with P(a)|n, then the Jesmanowicz' conjecture is true.(2) with a prime with p=-1 (mod 4). If n is a positive integer with P(n)|a, then the Jesmanowicz' conjecture is true.(3) where both p>3 and q>3 are distinct primes, and c is a positive integer with then the Jesmanowicz' conjecture is true.(4) If c?100, then the Jesmanowicz'conjecture is true.