Even integers as sums and differences of two primes

We use the Second Principle of Mathematical Induction to prove that every even integer greater than 2 can be expressed as the sum of two primes. The proof proceeds by first establishing, from the inductive hypothesis, that every even integer can be represented as the difference of two primes belonging to a finite set determined by the hypothesis. This result is then used to establish the inductive step for the representation as a sum of two primes. As an extension of this work, we prove that every even integer admits multiple representations as the difference of two primes.