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

• I
• Saracen Rue
In summary, the conversation discusses the functions ##f(x)## and ##g(x)##, where ##f(x)## is the power tower function and ##g(x)## is the primorial function. The point of intersection between these two functions is defined as ##(m, f(m))##, and the conversation asks for the value of ##m## and ##f(m)## to five significant figures. Additionally, there is a question about the definition of ##g(x)## for values of ##x## between ##e^{-e}## and ##e^{1/e}##, as well as how the primorial is defined for non-integers. The conversation also mentions using curve fitting to estimate the values.
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.

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?

How do you define the primorial for non-integers?

TeethWhitener said:
How do you define the primorial for non-integers?
Im looking for the best estimation using a curve fitting.

## What is an infinite power tower?

An infinite power tower is a mathematical concept where a number is raised to the power of itself, and then that result is raised to the power of itself, and so on, infinitely.

## What is a primorial?

A primorial is a product of the first n prime numbers, denoted as n#. For example, 5# = 2 x 3 x 5 = 30.

## What is the intersection point of an infinite power tower and a primorial?

The intersection point is the value where the infinite power tower and the primorial are equal.

## Is there a formula for finding the intersection point?

Yes, there is a formula for finding the intersection point of an infinite power tower and a primorial. It is given by the expression x = W(log(p)), where W is the Lambert W function and p is the primorial.

## Can the intersection point be calculated for any values of n and p?

Yes, the intersection point can be calculated for any positive integer values of n and p. However, for very large values, it may be computationally intensive to find the exact value.

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