The last theorem of Fermat has been solved in
recent years so I hear. He must have been a
smart fellow to know of the future
mathematical techniques needed to prove it.
Or he was kidding in that margin entry.
Or he was mistaken in his insight.
Or he had a simpler proof than has just been
promugated and it is still to be discovered.
I am not going to give any proof myself but I
was thinking about it many years ago and tried
to visualise the subject. It might be
original though I doubt it.
Think of a family of all right angled triangles
on a base with the right side the opposite and
the hypotenuse coming from the left end of the
base. As the opposite gets higher the angle
goes from small to 45 degrees. This family
includes all possible integer solutions at
various scales. After 45 degrees the family is
being repeated so that it is superfluous to
consider them after 45 degrees. You may have
the simple 3,4,5 example or nearby ones with
sides of 20, 30, 1000 or more integers as
scaled. Indeed though you can specify the
irational 1,1,root2 triangle, you can get
ever close to it with sides of hundred of
integers long scaled. But lets not go there!
Now the tip of the opposite side is in effect
the "loci" of all possible right angled
triangles. Now think of all possible triangles
that obey the cubed rule. The base is still
one of the smaller sides, the longest side is
still as with the hypotenuse of the right
angled, with the opposite sloping inwards.
The loci of these cubed triangles now becomes
an "s" shaped line going ever upwards. And
extends beyond 45 degrees. Loci for higher
powers are slightly straighter lines until
the power of infinity which merges with the
right angled triangle loci straight up.
If you have managed to follow so far, you know
there are no integer solutions on the curved
loci of cubed and above loci. But why? Does
this give any insight?