Horse 2813 - I Do Not Think That π^π^π^π Is An Integer

There was an interesting conjecture which Matt Parker, owner of the YouTube channel Stand Up Maths posed, which is that π^π^π^π could be an integer. I went through the range of emotions which he describes in the video in relation to this conjecture and have come to the conclusion that although I do not possess the mathematical ability to prove or disprove the conjecture, after fighting with it for more than a bit of time I have come to my own private position that I think that it is probably not an integer.

For the purpose of this post, I am going to assume that you already know some things about maths and if you don't, then it is a good idea to watch that video first.

Link:

Before we get to why I think that π^π^π^π is not an integer, there are some other important concepts to consider.

The first thing to notice about addition is that it is commutative, that is that the order in which you do things produces the same result. That's a very different statement than saying that the order doesn't matter because it very much does; for reasons that I won't go into here. Multiplication is also commutative. Multiplication however, is just fast addition. Likewise, raising something to a power and using an index notation, is just fast multiplication.

2⁵ = 2x2x2x2x2 or 2^5

π^π just means that you multiply π by itself π times. That's not a concept that we can wrap our heads around intuitively but since we can describe this using the rules of logic, then the nature of reality starts to look like pure objective fantasy. 

π^π^π^π means that pi is being raised to the power of pi, then that is being raised to the power of pi, then that is being raised to the power of pi.

To give you an idea of how massive that number is, 2^2^2^2 is 65,536. 3^3^3^3 is a number so massive that I can not calculate it sensibly. 3^3^3 = 7,625,597,484,987 which is a bit big. That means that 3^3^3^3 is 3 multiplied by itself 7,625,597,484,987 times. π^π^π^π is at least more massive than that and since π^π^π is a number that is about 1,340,164,183,006,357,435, then mutiplying π by itself is even less sensible.

Of huge importance here, is that, index notation is not commutative and requires you to perform various operations from the top of the power tower to the bottom of the power tower before you arrive at a final result. That's a problem.

Unlike the other examples given in the video, such as √2 being multiplied by itself to give 2, or e being the result of an inverse function of natural logarithms, π is none of those things. π is a relatively small number which has a value of 3 and a bit but isn't a rational number (that is one that can be expressed as a/b, or the ratio of a to b, hence rational) nor is it the result of some function which even has an inverse function.

What a/b^a/b tells you is that we are looking for the bth root of a/b^a. Since we know that π is irrational, then either a or b or both must also be irrational. That's a different set of circumstances to playing with √2 because √2 is a fancy way of writing 2^-1. -1 is rational. In fact, while √2 is irrational, all of the various components which define it in exponential notation are rational. It is possible that that might be true for π but given that we've had centuries thinking about the problem, it looks unlikely.

Intuitively, because π can not be written in the form of a/b then the power tower of a/b^a/b^a/b^a/b isn't going to spit out something rational because either a or b if a/b=π can not be rational and possibly neither of them can be. Then what?

There isn't anything that I can see that would give rise to π^π^π^π being rational because I can test π^π which is one of the component parts of the whole. Now by itself that doesn't necessarily mean that the whole thing is also irrational but unless π^π^π undoes whatever π by itself is doing, then I just don't see it.