For any given coprime integers p and q greater than 1, in 1959, B proved that all sufficiently large integers can be expressed as a sum of pairwise terms of the form p^aq^b. As Davenport observed, Birch＇s proof can be modified that the exponent b can be bounded in terms of p and q. In 2000, N. Hegyvari effective version of this bound. The author improves this bound.
Let l1,l2,…,lg be even integers and x be a sufficiently large number.In this paper,the authors prove that the number of positive odd integers k≤x such that（k+l1）2,（k+l2）2,…,（k+lg）2 can not be expressed as 2n+pαis at least c（g）x,where p is an odd prime and the constant c（g）depends only on g.
Consider all the arithmetic progressions of odd numbers, no term of which is of the form 2^k ＋ p, where k is a positive integer and p is an odd prime. ErdSs ever asked whether all these progressions can be obtained from covering congruences. In this paper, we characterize all arithmetic progressions in which there are positive proportion natural numbers that can be expressed in the form 2^k ＋ p, and give a quantitative form of Romanoff＇s theorem on arithmetic progressions. As a corollary, we prove that the answer to the above Erdos problem is affirmative.
