Minus One But What a Difference
Prove that for each integer n ≥ 3 the equation
x n + y n = z n-1
has infinitely many solutions in positive integers.
References
- T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations, Birkhäuser, 2010, p. 23
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Just verify that
x = k(kn + 1)n-2,
y = (kn + 1)n-2, and
z = (kn + 1)n-1
y = (kn + 1)n-2, and
z = (kn + 1)n-1
solve the equation. Indeed,
[k(kn + 1)n-2]n + [(kn + 1)n-2]n =
(kn + 1)n(n-2) + 1 = (kn + 1)(n-1)²
while also
[(kn + 1)n-1]n-1 = (kn + 1)(n-1)².
Why does not this work for the FLT? Just minus 1 - but what a difference!
An even stranger example has been furnished by Emmanuel Moreau from France. Set
x = kn-1 · 2n-2,
y = x,
z = kn · 2n-1.
y = x,
z = kn · 2n-1.
and verify that xn + yn = zn-1. For the original FLT equation one would not dream of having
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