Prove No Solution for Equation $y^3=1+2^{2^x}$ in Natural Numbers

[anemone]

Prove that the equation $\large y^3=1+2^{2^x}$has no solution in natural numbers.

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[anemone]

Congratulations to the following member for his correct solution:

1. greg1313

Solution from greg1313:

Spoiler

Proof by induction.

Base case: $$1^3-1=0\ne2^{2^x}$$

Inductive step: $$(n+1)^3-1=n(n^2+3n+3)=2^{2^x}$$$n^2+3n+3$ is odd, so it cannot divide $2^{2^x}$.

We have therefore proved from induction method that the equation $\large y^3=1+2^{2^x}$ has no solution.