# 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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