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.

Solution

References

  1. T. Andreescu, D. Andrica, I. Cucurezeanu, An Introduction to Diophantine Equations, Birkhäuser, 2010, p. 23

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Copyright © 1996-2017 Alexander Bogomolny

Just verify that

x = k(kn + 1)n-2,
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.

and verify that xn + yn = zn-1. For the original FLT equation one would not dream of having x = y.


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