# Find the intersection point of an infinite power tower and a primorial

• I
• Saracen Rue
Saracen Rue
TL;DR Summary
Find the point of intersection between ##f(x) = {^{\infty}x} = x \uparrow \uparrow \infty## and ##g(x)=p_{x}###
Consider ##f(x) = {^{\infty}x} = x \uparrow \uparrow \infty## and ##g(x)=p_{x}###, where ##p_x### is the primorial function and is defined such that ##p_n### is the product of the first ##n## prime numbers. For example, ##p_{4}### ##= 2×3×5×7=210##

Let the point of intersection be defined as ##(m, f(m))##; determine the value of both ##m## and ##f(m)## correct to five significant figures.

mitochan
I think power tower is defined and finite for $$e^{-e}<x<e^{1/e}$$. How g(x) is defined for x in this region?