A proof with a mistake ( or not )

Fermat's Last Theorem is a theorem that was proved only after 358 years. The theorem states that for all $n \geq 3$ , the equation $a^{n} + b^{n} = c^{n}$ has no positive integer solutions.Below is a proof for the special case $n = 3$ .Given that $c - a < b$ (which is easy to prove),find the mistake.

Step 1 - $a^{3} + b^{3} < a^{3} + (c-a)^{3}$ because $c - a < b$ .

Step 2 - $a^{3} + b^{3} < c^{3} - 3ac^{2} + 3a^{2}c$ after calculating the value of $(c-a)^{3}$ and removing the $a^{3}$ terms.

Step 3 - $c^{3} < c^{3} - 3ac(c-a)$ by replacing $a^{3} + b^{3}$ with $c^{3}$ and factoring.But this implies that $c^{3}$ is smaller than something smaller than itself,so $a^{3} + b^{3} \neq c^{3}$ .

• If you like this problem,try Part 2 !