Stilkol's equation

Stilkol's equation, also called the Stilkol's product equation, is of the form $x^{n}=n^{x},$ where n is a given positive integer , bigger than 1

The conjecture is:

When n is an odd integer there are two real solutions to the equation

When n is an even integer there are three real solutions to the equation

History

The equation was presented and examined by Ronen Stilkol at year 1980 when he was at his high school studies The first equation tested from this series was $x^{2}=2^{x},$ here two solutions: 2 & 4 are noticeable and a third solution exist at -0.767

The equation was used by Mr. Stilkol in an acceptance exam test for engineer's as exploring the various solutions can

give information about the method used in order to solve the equation and the capability of solving it.

Since presenting the equation to engineering job candidates the equation was sent on the internet and there are various solutions

suggested for the $x^{2}=2^{x},$ form from the series, some use lambert W function , to get the negative solution.

References

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