Let pi be prime numbers. In this paper, it is proved that for any integer k≧5, with at most exceptions, all positive even integers up to N can be expressed in the form . This improves the result for some c>0 due to Lu and Shan , and it is a generalization for a series of results of Ren and Tsang ,  and Bauer [1–4] for the problem in the form . This method can also be used for some other similar forms.
 Bauer, C.1998On a problem of the Goldbach–Waring typeActa Math. Sinica, New Series14223–234.