ox_n

Consecutive base numbers will necessarily alternate between even and odd. So even the closest base numbers still have a gap between their resulting odd number series, which only increases as the distance between base numbers increases.

Yes.

x and y will be a multiple of the base number.

What other mindset do you see working better?

_DON'T DOWN VOTE JUST BECAUSE YOU CAN'T DO MATH_

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:

    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.

It's not that hard people. Stop believing everything you're told about how "hard" something is.

HINT: The number of odd numbers in the series exactly matches the starting square base number

It doesn't work in my browser. Shame. I really felt like wasting some time just now.

I saw a copyright on every version of that image I could find.

I'm real curious how he got the rights to the image on the cover.