Final answer:
Using properties of cubes and algebraic manipulation, it's shown that there are no integer solutions for the equations m³ - n³ = 3 and m³ - n³ = 4 due to the fact that the differences in cubes of consecutive integers are larger than 3 and 4.
Step-by-step explanation:
To show that there are no integer solutions to the equations m³ − n³ = 3 and m³ − n³ = 4, we can use the properties of cubes of integers. For any two integers m and n, the difference m³ - n³ is also an integer. When considering two consecutive integers, their cubes have a difference that is larger than 3 or 4.
Let's consider the first equation m³ - n³ = 3. If n = m - 1, expanding the cubes gives us m³ - (m-1)³, which simplifies to 3m² - 3m + 1. This expression can never be equal to 3 for any integer value of m.
Now, let's consider the second equation m³ - n³ = 4. Similarly, substituting n = m - 1 and expanding the cubes results in 3m² - 3m + 1, which can't equal 4 either. Additionally, for integer solutions, there's always an integer k such that n = m - k. The difference between their cubes will grow significantly larger than 4 as m increases. Thus, no integers m and n, where m > n, can satisfy m³ - n³ = 4.
Therefore, we conclude that there are no integer solutions for both equations.