Fermat's Like Equaition
Here's a problem proposed by Albert F. S. Wong (Mathematics Magazine, Vol. 77, No. 3 (Jun., 2004)):
For which positive integers k does the equation
X2k-1 + y2k = z2k+l
have a solution in positive integers x, y, and z?
The solution is by Jerry W. Grossman (Mathematics Magazine, Vol. 78, No. 3, (Jun., 2005)).
There are solutions for all k. Because 2k - 1, 2k, and 2k + 1 are pairwise relatively prime, it follows from the Chinese Remainder Theorem that there is a positive integer m with
m ≡ 0 (mod 2k),
m ≡ 0 (mod 2k + 1),
m ≡ -1(mod 2k - 1).
There are then positive integers r, s, t with
m = r(2k) = s(2k + 1) = t(2k - 1) - 1.
Now let a = 32k+1 - 22k, so a + 22k = 32k+1. Multiply through by am to obtain
am+1 + am22k = am32k+1.
This can be put into the form
(at)2k-1 + (2ar)2k = (3as)2k+1,
giving a solution to the diophantine equation.
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