Epilogue: More Curves and Fermat

By Michael Griffin, MLS

An alternative statement of Fermat's theorem is that any solutions would have to have coordinates on the Desmos curves (shown in the Prologue) that could be converted into natural number triples. This does happen on the unit circle for n = 2. Note the two coordinate points (0.6, 0.8) and (0.8, 0.6).  As fractions these are (3/5, 4/5) and (4/5, 3/5) which convert into the triple  345. Another result occurs below the baseline using fractional values of n. For example the square root is n = 1/2

 

Zooming into the first quadrant for this we can see that the point (0.25, 0.25) has a solution (0.5, 0.5) on the n = 1 baseline which would convert into the triple 112

That is not quite fitting the rules since ABC are supposed to be distinct. The points on the baseline (0.25, 0.75) and (0.75, 0.25) would become distinct as 134 or 314. Still we might generalize Fermat's theorem as that any natural number solutions would fit the condition of n between or equal to 1/2 and 2, or 1/2 < n < 2. Higher roots would have the same unsolvable problem as higher powers. Or at least intuitively they seem like they should. And the principle would hold that any convertible solution points on the square root curve would never occur on the higher roots curves. However, this is getting away from the blog posts which  only dealt with natural numbered powers, not roots.

Addendum

Getting back to the previous blog post about the nth root we may note that there is a valid general principle that any nth root must be smaller than a Euclidean square root. If we then look at the first graph of Desmos curves we see that all of the higher power curves are beyond the second power curve and thus their coordinates are larger than the coordinates of the second power curve. This seems to contradict the principle that higher roots would be smaller than square roots and could thus be mistaken as a graphical proof of Fermat’s Last Theorem. Of course when the decimal coordinates are converted into fractions and then the fractions are converted into triples we would then get smaller roots than the second power, fitting the notion that there are infinitely many solutions for any nth power curve, just none that convert into natural numbers for all three elements of any triple. Which remains to be proved.

Sources

https://www.desmos.com/calculator