_DON'T DOWN VOTE JUST BECAUSE YOU CAN'T DO MATH_<p>The proof Fermat hinted to was about the difference between squares. All whole numbers taken to a power greater than two (n^3) can be represented as the difference between two whole squares (x^2 - y^2). These differences can then be shown as the sum of consecutive odd numbers:<p><pre><code> 2^3 = 3^2 - 1^2 = (1+3+5) - (1) = 8,
3^3 = 6^2 - 3^2 = (1+3+5+7+9+11) - (1+3+5) = 27,
4^3 = 10^2 - 6^2 = (1+3+5+7+9+11+13+15+17+19) - (1+3+5+7+9+11) = 64
5^3 = 15^2 - 10^2 = (21+23+25+27+29) = 125
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When you examine the odd number series that results from each base, you'll discover that there will always be a gap if you try and combine two odd number series together, which explains Fermat's little joke about margins. The same trick works for higher powers.<p>It's not that hard people. Stop believing everything you're told about how "hard" something is.<p><i></i><i>HINT:</i><i></i>
The number of odd numbers in the series exactly matches the starting square base number