On The Sum Of A Prime Power And Two Powers Of An Integer

In this paper, based on P.Erdos' problem and on the work of Yong-Gao Chen, we have proved that for any odd integer a which is not a power of 2 minus 1, there are infinitely many positive odd integers M, (M, a) = 1 and (M - 2, a - 1) = 1, which cannot be written in the form a~k + a~l + p~α, where k, l,αare nonnegative integers and p is a prime integer.We have used two different methods to prove our main result. The principal way is both covering system which is defined by P.Erdos and the Chinese remainder theorem. We have constructed a covering system with distinct modulus and without powers of 2 as modulus. And we also found a lemma which enable us not only to succeed to almost all the odd integer a from a fixed odd integer a as well as a = 5, 9, 11; but also to simplify much the proof.We have also proved that the theorem is correspondingly true when a is some even numbers. For example, a = 6, 10, 12. For the other even numbers a, we can treat them with the same method, which needs much complicated calculation. So, the work continues...