Can we use Peano's axioms to prove that integer = prime + integer? Every integer greater than 2 can be expressed as sum of some prime number greater than 2 and some nonegative integer....$n=p+m$. Since 3=3+0; 4=3+1; 5=3+2 or 5=5+0...etc it is obvious that statement is true.My question is: Can we use Peano's axioms to prove this statement (especially sixth axiom which states "For every natural number $n$, $S(n)$ is a natural number.")?

Yes we can use the Peano's axiom to prove that integer = prime + integer. Think of $0 + a = a$. 
