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
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.