There are infinitely many odd numbers not expressible as the sum of a prime number and a power of $2$ Prove that there are infinitely many odd integers that are not expressible as the sum of a prime number and a power of two.
This is a difficult problem. Please give me some hints and some examples of this.

Erdos proved this in the paper in which he introduced the concept of covering congruences. The reference is On integers of the form $2^n+ p$  and some related problems, Summa Brasil Math 11(1950), 1-11. The proof is given by Fabrykowski and Smotzer, Covering systems of congruences, Math Mag 78 (2005) 228-231, available at http://www.maa.org/sites/default/files/3004416309960.pdf.bannered.pdf
EDIT. The Erdos paper is available at http://www.renyi.hu/~p_erdos/1950-07.pdf. Some of it is heavy going, but the part that answers the question here is Theorem 3 and its proof, which is quite elementary. http://www.renyi.hu/~p_erdos/ has the collected papers of Erdos. 
