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

Solution

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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

x = ubm(uabm + vabm)bm and
y = vam(uabm + vabm)am.

Then

xa + yb = (uabm + vabm)abm + 1 = (uabm + vabm)cn.

So letting z = (uabm + vabm)n we get a 2-parameter solution to (3).


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|Contact| |Front page| |Contents| |Algebra| |Inventor's Paradox|

Copyright © 1996-2018 Alexander Bogomolny

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