Power Equation

Consider the equation

(1)

for . Taking natural logarithms gives

(2)

Besides the diagonal solutions , there is exactly one solution for each , and exactly one solution for each . The distinct solution is

(3)

where is the Lambert W-function. This function is illustrated above by the blue curve. For or , only the diagonal solution exists.

Writing gives , so all the positive real solutions with have the parameterization

			(4)
			(5)

where and (Bennett and Reznick 2004, pp. 13-14). Replacing by interchanges and . As , both approach . This parameterization also occurs in Trott (2006).

All positive rational solutions with are given by

			(6)
			(7)

for integers or (Dunn 1980, Pickover 2002; Bennett and Reznick 2004, p. 14). These solutions are shown on the plot as red dots. For positive integer , gives , while gives the same pair in reverse order. The values and are excluded because the formulas are undefined there. The only positive integer solutions with are and (Bennett and Reznick 2004, p. 14).