Numbers

For two natural numbers m &n prove that m³ + n³ + 4 cannot be a perfect cube

Answer:

Let, m³ + n³ + 4 be a perfect cube

Now, (m + n)³ = m³ + n³ + 3mn(m +n)

Comparing, 3mn(m + n) = 4

Or, mn(m + n) = 4/3

Since m and n are elements of natural number, so, both mn and (m + n) are natural numbers and therefore there product cannot be a fraction.

So, the hypothesis that m³ + n³ + 4 be a perfect cube is wrong.

So, m³ + n³ + 4 can never be a perfect square if n, m are elements of Natural number.