# 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

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). • Diophantine Equations
• Finicky Diophantine Equations
• Diophantine Quadratic Equation in Three Variables
• An Equation in Reciprocals
• A Short Equation in Reciprocals
• Minus One But What a Difference
• Chinese Remainder Theorem
• Step into the Elliptic Realm
• Fermat's Like Equation
• Sylvester's Problem
• Sylvester's Problem, A Second Look
• Negative Coconuts
• 