
(pi^4+pi^5)^(1/6) - DavidSJ
https://www.google.com/search?q=(pi%5E4%2Bpi%5E5)%5E(1%2F6)
======
ColinWright
Equivalently:

[https://www.google.com/search?q=%28pi^4%2Bpi^5%29^%28-1%2F6%...](https://www.google.com/search?q=%28pi^4%2Bpi^5%29^%28-1%2F6%29*e)

Or:

[https://www.google.com/search?q=e^6%2F%28pi^4%2Bpi^5%29](https://www.google.com/search?q=e^6%2F%28pi^4%2Bpi^5%29)

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](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[0][1], and you end up with more
coincidences than you might expect.

[0]
[https://news.ycombinator.com/item?id=1312636](https://news.ycombinator.com/item?id=1312636)

[1]
[https://news.ycombinator.com/item?id=4753014](https://news.ycombinator.com/item?id=4753014)

~~~
3rd3
On the other hand there are infinitely many small numbers. Maybe one could
express it more precisely using Kolmogorov complexity?

[http://en.wikipedia.org/wiki/Kolmogorov_complexity](http://en.wikipedia.org/wiki/Kolmogorov_complexity)

~~~
3rd3
Why the downvotes? Do you mean natural numbers?

------
eps
Pfft ... this is _nothing_ compared to

    
    
      1.49 ^ 2.87
    

yielding Pi, whereby 1.49 is a price of a 250cl bottle of vodka and 2.87 is
the price of 0.5l of the same. Former is known as "chekushka" and latter - as
"pol-litra", both are _the_ staples of Soviet Union alcohol landscape with
their prices remaining stable for decades and permanently etched into the
heads of every Russian, drinking or not.

~~~
raldi
I remember reading somewhere that in Russian culture the optimal number of
people to drink together is customarily considered to be three, because they
could each toss in a ruble and that would pay for a half-liter of vodka to
share and a small snack.

------
cies
I think Eurler's identity is far more "interesting":

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 :)

[http://en.wikipedia.org/wiki/Euler's_identity](http://en.wikipedia.org/wiki/Euler's_identity)

~~~
mijoharas
One thing I've never understood is why people write it that way round, I
always think of it as:

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.

~~~
waynecochran
Because it involves all of the most important a constants: 0, 1, e, pi, i.

------
jamesrom
Brief explanation for those who aren't sure what this is about.

(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.

~~~
GregBuchholz
Is there an theorem that says that you can't compute e from pi? Maybe this is
just the second term/iteration of that algorithm. I'd be interested in seeing
other formulas relating pi and e.

~~~
nhaehnle
A related mathematical concept is that of _algebraic independence_. Two
numbers x and y are algebraically dependent (over the rationals) if there is a
rational bivariate polynomial p such that p(x,y) = 0. As far as I know, pi and
e are not known to be algebraically independent, though I guess most people
would expect them to be.

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).

~~~
tgb
Wow, as someone who is studying math I'm surprised to hear that that isn't
known. Wikipedia even says that it's not known whether e + pi is irrational.

------
AliAdams
I'm feeling a bit ignorant.

Any math guys want to elaborate on the relevance of the result?

~~~
ColinWright
What is there about my comment[0] that you feel is lacking? I'd be happy to
elaborate further, but I'm not sure what you think I've missed out.

[0]
[https://news.ycombinator.com/item?id=7892425](https://news.ycombinator.com/item?id=7892425)

------
RivieraKid
I wish people didn't upvote low quality submissions like this.

~~~
ColinWright
While it's true that the submission itself is bare and unadorned, the topic is
an interesting one. It would be nice if someone had actually taken the time to
write a proper blog post and explained and expanded on the points, but that's
happening here in this thread.

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.

~~~
okamiueru
I think the reason that it is of questionable interest and value, is the same
for why there is no proper blog post explaining it.

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](http://en.wikipedia.org/wiki/Mathematical_coincidence)

------
jmount
Numerical near-miss coincidences are to be expected when working with
transcendental numbers. First reason is the pigeon hole observation others
have made. The other reason is: this is one of the properties of
transcendental numbers. Roughly: one technique used in showing a number to be
transcendental is you show it is not equal to any term in sequence of
algebraic numbers and yet closer to them than (without equality) than any
algebraic number could be.

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.

------
lucb1e
Am I missing something on mobile? I only see the result is 2.x and then google
results that don't seem relevant.

~~~
puzzlingcaptcha
[http://en.wikipedia.org/wiki/E_%28mathematical_constant%29](http://en.wikipedia.org/wiki/E_%28mathematical_constant%29)
&
[http://en.wikipedia.org/wiki/Mathematical_coincidence](http://en.wikipedia.org/wiki/Mathematical_coincidence)

------
thegeomaster
Ah. Reminds me of [http://xkcd.com/217](http://xkcd.com/217) :)

~~~
ColinWright
Related: [http://xkcd.com/1047/](http://xkcd.com/1047/)

------
aylons
And 10! seconds is exactly 42 days. Not an approximation.

~~~
dtech
That's not _really_ a fundamental coincidence, but a result of the mechanics
of factorial and our time system.

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

~~~
zhte415
42 is not a mathematical constance?

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?

------
6UwUFxLt
[http://mathworld.wolfram.com/AlmostInteger.html](http://mathworld.wolfram.com/AlmostInteger.html)

------
zatkin
Another coincidence with pi and phi (golden ratio):
[https://www.google.com/search?q=4%2Fsqrt(phi)](https://www.google.com/search?q=4%2Fsqrt\(phi\))

------
yammesicka
Cool! Also:
[https://en.wikipedia.org/wiki/Mathematical_coincidence](https://en.wikipedia.org/wiki/Mathematical_coincidence)

------
davyjones
i th root of i

Not what you would expect.

~~~
3rd3
Which is the same as i^(i^(i^i))).

