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This is not an answer, but the solution to $x^{x+1}=(x+1)^{x}$ is known as the first Foias Constant, FC = 2.2931662874118610315.....
This is not an answer, but the solution to $x^{x+1}=(x+1)^{x}$ is known as the first Foias Constant, FC = 2.2931662874118610315.....