1. **Problem Statement:** Prove that the difference between the squares of any two consecutive integers is equal to the sum of these integers. Then, find two consecutive integers whose squares differ by 259. 2. **Proof:** Let the two consecutive integers be $n$ and $n+1$. The difference between their squares is: $$ (n+1)^2 - n^2 $$ 3. Expand each square: $$ (n^2 + 2n + 1) - n^2 = 2n + 1 $$ 4. Notice that the sum of the two integers is: $$ n + (n+1) = 2n + 1 $$ 5. Therefore, $$ (n+1)^2 - n^2 = n + (n+1) $$ which proves the statement. 6. **Finding the integers with difference of squares equal to 259:** We have from the proof: $$ 2n + 1 = 259 $$ 7. Solve for $n$: $$ 2n = 259 - 1 $$ $$ 2n = 258 $$ $$ n = 129 $$ 8. The two consecutive integers are 129 and 130. 9. Verify by calculating the difference of their squares: $$ 130^2 - 129^2 = (130 + 129)(130 - 129) = 259 \times 1 = 259 $$ Thus, the two consecutive integers are $129$ and $130$.