The Solvability For The Equation φ_e（n）=2^ω（n）（e=5,8,12）

In the 18th century,Euler defined the Euler function φ（n）for a positive integer n,which plays an important role in discrete mathematics,information security,and so on.For example,the Euler function plays a key role in RSA public-key cryptosystem established by R.Rivest,A.Shamir and L.Adelman.In 2007,for the positive integers n and e,Cai,et al,generalized the definition for the Euler function to be the generalized Euler function φe（n）.Especially,φ1（n）=φ（n）.So far,only for the case e∈｛1,2,3,4,5,6,8,12},the explicit formula for φe（n）has been determined.Based on the explicit formula for φe（n）（e=1,2,3,4,6）,all positive integer solutions for the equation φe（n）=2ω（n）（e=1,2,3,4,6）have been determined.Basing on the explict formula for φe（n）（e=5,8,12）,elementary methods and techniques,we discuss the solvability for the equation φe（n）=2ω（n）（e=5,8,12）,and then their positive integer solutions are determined completely.