Prologue: Curves and Fermat
By Michael Griffin, MLS
In Fermat's Last Theorem, An + Bn = Cn has infinitely many solutions for n but the highest whole number solution is 2, as with the sides of a right triangle in the Pythagorean theorem. These solutions are called Pythagorean triples; for example the first natural numbers triple ABC is 345 since 9 + 16 = 25. Although it has been solved by modern methods, Fermat's Last Theorem will continue to draw anyone curious as to a method Fermat himself might have used to solve it. To that end, a few idle investigations are reported here in future posts, with some conclusions that by no means make any proofs except on the set of Pythagorean triples. While these results might make for some interesting algebra, they are so trivial that it is not surprising if no one has published them before. To show just how trivial they are, we will look at some graphs that show the Captain Obvious nature of why any Pythagorean triple would not work at a higher power.
We begin with a graph from an online textbook (schoolbag) which gives a geometric set-up for a proof of Euclid's formula to generate Pythagorean triples:
What matters to us is that the general Pythagorean theorem is converted Into a unit circle here by division:
If A2 + B2 = C2 then (A2 + B2 = C2) / C2 and A2 / C2 + B2 / C2 = C2 / C2
and A2 / C2 + B2 / C2 = 1 so we define A2 / C2 = X2 and B2 / C2 = Y2
So we have a unit circle of X2 + Y2 = 1 shown in the first quadrant above.
The natural number solutions for Pythagorean triples will lie on the unit circle.
It is easy enough to show that these points will stay on the unit circle and not appear in higher curves.
With the same kind of division process we can turn higher powers into unit equations and graph them. So A3 + B3 = C3 becomes X3 + Y3 = 1, A4 + B4 = C4 becomes X4 + Y4 = 1, A5 + B5 = C5 becomes X5 + Y5 = 1, etc. Using the online Desmos graphing calculator we can show these equations and their curves. Even powers give closed figures starting with a circle n = 2 whereas odd powers have unbounded lines that in the first quadrant curve out near the even powers.
We zoom in on the first quadrant because it would be the positive values to fit Fermat's theorem. Zooming in also shows that as n increases the curvature moves further out from the circle in an orderly progression. The straight line is our beginning baseline when n = 1 so
X + Y= 1.
The curves never intersect except at the points (0,1) and (1,0). So any natural number Pythagorean triples never happen on higher power curves, which is a visual depiction that these triples would never provide solutions to Fermat's Last Theorem. All of these graphs simply serve as alternative proofs of the algebra to come.
Sources
https://www.desmos.com/calculator
EUCLID'S METHOD FOR FINDING PYTHAGOREAN TRIPLES
https://schoolbag.info/mathematics/numbers/79.html
