Question

Prove that there do not exist positive integers m and n such that m² - n² = 1?

Answer 1

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.