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Edited: 13/01/2015 04:10 PM
Picture: John Rowland
John Rowland
Crazy curves
 
Except for the circle (which is shown for reference), on the other three curves, except where the curves cross an axis, all the points (x,y) have an interesting property. In fact this property holds for all the curves of this family for any power of x and y above 2. What is so special about the co-ordinates (x,y) of these points and why?
Edited: 13/01/2015 04:10 PM
Picture: John Rowland
John Rowland

The remarkable property of (x,y) is that it is not possible for both x and y to be rational numbers. If this were so then x=a/b and y=c/d implies (ad)n + (cb)n = (bd)n where n>=3 and ad, cb and bd are all integer - which is impossible by Fermat's Last Theorem.