Well since Animal's calculator apparently needs new C batteries (
), here's a few hints:
1) the pattern is *not* that the first series' lines are all the positive integers, with the beginning digit of the next line being 1 plus the digit ending the previous line. It's much more subtle than that. The squared series is actually related to the first series.
2) nope, figure it out yourself.
3) obviously not, that's the gist of Fermat's Last Theorem, proved a while back finally.
4) The first two series hold because it's possible to cut up a number on the left side of each equation, then "wrap" the cut parts around the uncut numbers until they come out equal to the right side. That only holds for equations of exponent 1 and 2. The wrapping does not work on higher exponents because it's not possible to cut up any of the numbers on the left side to make equal equations. The cut parts don't add up enough to wrap.
But it
is possible to "complete the square" (essentially add some bits and pieces) for higher powers that cause Fermat's Last Theorem to hold for non-integer higher powers, just not with integers. However, there is no pattern that holds for every higher power. The non-integer working power will be either a little bigger or a little smaller than each non-working integer higher exponent. But if there is any rhyme or reason for every power series, it's not at all apparent. (fixed wording a bit)
