# A proof with a mistake ( or not )

**Algebra**Level pending

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!

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