Two-Parameter Solutions to Three Almost Fermat Equations
The problem below has been proposed by Morrey Klamkin (Crux Mathematicorum, 1960):
Determine two parameter solutions of the following "almost" Fermat Diophantine equations:
(1) | x n-1 + y n-1 | = z n |
(2) | x n+1 + y n+1 | = z n |
(3) | x n+1 + y n-1 | = z n |
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Copyright © 1996-2018 Alexander Bogomolny
Determine two parameter solutions of the following "almost" Fermat Diophantine equations:
(1) | x n-1 + y n-1 | = z n |
(2) | x n+1 + y n+1 | = z n |
(3) | x n+1 + y n-1 | = z n |
The solution is by Leo Moser (Crux Mathematicorum, September 1961, #456).
We will exhibit two parameter solutions for a more general equation:
(3) | xa + yb = zc, (a, b, c) = 1. |
Since (a, b, c) = 1, we can first find m and n such that
(4) | abm + 1 = cn. |
Now let
y = vam(uabm + vabm)am.
Then
So letting z = (uabm + vabm)n we get a 2-parameter solution to (3).

|Contact| |Front page| |Contents| |Algebra| |Inventor's Paradox|
Copyright © 1996-2018 Alexander Bogomolny
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