The point being, of course, that pi^4 + pi^5 is very, very nearly e^6.
See also: https://news.ycombinator.com/item?id=7892430
The idea is that there are only so many small numbers, and there are lots of ways of combining things together. The Pigeonhole Principle then says that if you stuff too many things into a small enough space, some of them will be close together. Although apparently obvious, this is more widely applicable than people generally realize. It's used, for example, in one of the proofs that that every prime of the form 4k+1 is expressible as the sum of two squares (Examples: 29=4x7+1=5^2+2^2, 181=4x45+1=10^2+9^2, 193=4x48+1=12^2+7^2).
Combine this with the birthday problem/paradox, and you end up with more coincidences than you might expect.
pi^4 + pi^5 - e^6 = -0.00001767345...
1.49 ^ 2.87
e^(i * pi) + 1 = 0
It also connects "e" and "pi"; but on top of that also "i", "1" and "0". When I first read this equation it felt like a proof of God's existence :)
e^(i*pi) = -1
I don't understand why people switch it around all the time, I don't think it loses any elegance going that way round, I also feel it loses clarity being shuffled around because it's just one more (admittedly tiny) operation you need to do to see why it happens.
(pi^4+pi^5)^(1/6) is an approximation of the mathematical constant e.
It's a noteworthy result, but nothing more than a curiosity. At first glance it may appear to show some relationship between these two separate constants, but it doesn't.
Simply put, there's infinite ways to construct an approximation of e with pi (and vice versa) and some small subset of these ways are bound to look elegant or simple.
On the other hand, it is true that there must be some (actually infinitely many) infinite series in x with rational coefficients such that plugging x = pi gives e in the limit, but in itself, that's a trivial statement (just build the coefficients of the series in such a way that you force an approximation of e). It is also not a very interesting statement, because e is the limit of a series more or less by definition (what I mean here is that plugging 1 into the series that defines the exponential function).
A much more interesting relationship is Euler's identity, e^(i pi) = -1. There are many more formulas in which e and pi appear together in more or less interesting ways once you start going deeper into mathematics, but they tend to be not of the form discussed in this submission (I'm partial to Stirling's approximation as the "next" step after Euler's identity).
Any math guys want to elaborate on the relevance of the result?
The way you've put it sounds like if mathematicians are trying to figure out e they are applying this equation (I'm of course giving assuming this is unintentional and the result of commenting quickly).
And it's more relevant to computing, algorithms, statistics, and big data analysis than most people realize. People have a tendency to see too much in coincidences, people don't realize that there will be things that turn out to be equal, or nearly equal, more-or-less "by accident."
So while you may think this is a low-quality submission, I think you are only half right, and I think it's worth having.
It's a coincidence found from playing with numbers. There are infinite "oh, neat, this simple expression is close to this other constant" coincidences, none of which provide any deeper understanding or appreciable use.
Here is the wikipedia entry closes fits, and might actually be of interest: http://en.wikipedia.org/wiki/Mathematical_coincidence
And this is something I never liked about a lot of "math puzzles." A lot of them point out some effect as surprising or esoteric when the effect is actually a specific example of something one should expect (given a developed mathematical intuition). In the end the observation is only hermetic or exotic to those who don't know math and ends up being a barrier to getting comfortable with known results and their consequences.
(pi^4+pi^5)^(1/6) is a coincidental approximation to the mathematical constant e
10! = (10 * 9 * 8 * 5) * (4 3 2) * (7 * 6) = 3600 * 24 * 42
3600 = 1 hour in seconds
24 = 1 day in hours
Waaay less a "coincidence" than strange (approximate) relationships between the fundamental constants of mathematics
It is the answer to the ultimate question of life, the universe, and everything.
This is demonstrated by 42 being: Represented by 101010 in binary; The refraction angle of light off water in the forming of a rainbow; Light requires 10^-42 seconds to cross a proton; as well as being the result of 6*9.
Or have I simply been misled?
Not what you would expect.