Final answer:
To prove that there do not exist positive integers m and n such that m² - n² = 1, we can use a proof by contradiction. Assuming that such integers exist, we can rewrite the equation as (m + n)(m - n) = 1. However, solving the system of equations gives us m = 1 and n = 0, which are not positive integers.
Step-by-step explanation:
To prove that there do not exist positive integers m and n such that m² - n² = 1, we can use a proof by contradiction. Assuming that such integers exist, we can rewrite the equation as (m + n)(m - n) = 1.
Since 1 is a prime number, the only possible factors are 1 and itself.
This means that both (m + n) and (m - n) must equal 1 or -1. However, solving the system of equations gives us m = 1 and n = 0, which are not positive integers. Therefore, our assumption is false, and there do not exist positive integers m and n that satisfy the equation.