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Question

Prove that there do not exist natural numbers x and y, with x>1, such that x71x1=y5+1.

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Solution

Simple factorization gives y5=x(x3+1)(x2+x+1).
The three factors on the right are mutually coprime and hence they all have to be fifth powers.
In particular, x=r5 for some integer r.
This implies x3+1=r15+1, which is not a fifth power unless r=1 or r=0.
there are no solutions to the given equation.

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