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(pi^4+pi^5)^(1/6) (google.com)
52 points by DavidSJ on June 14, 2014 | hide | past | web | favorite | 45 comments





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

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

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

    pi^4 + pi^5 - e^6 = -0.00001767345...

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


Why the downvotes? Do you mean natural numbers?

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.

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.

If you're going to memorize 6 digits, I'd hope you would get more than 6 digits of accuracy at approximating pi. Try 355/113

3.141 takes up less space and is more accurate.

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 :)


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.

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

It's true that this may be one of most startling equations, but its far from being the concoction of the arbitrary approximation in the OP. This is just the special case of Euler's more general formula e^{ix} = \cos{x} + i\sin{x}, which can easily be derived from Taylor series for e^x, sine, and cosine.

Of course Euler's identity is far more interesting. That's an actual mathematical insight. This is just a coincidence from playing around with irrational constants. Two entirely different things.

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.

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.

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).

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.

e^(i * pi) + 1 = 0

I'm feeling a bit ignorant.

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

It's a mathematical equivalent of Virgin Mary on a toast. Two expressions not related by any close theory yield almost the same result.

This is the missing link: http://mathworld.wolfram.com/e.html (Compare the result of the expression to the value of the mathematical constant e. They coincide approximately [1].

[1] http://en.wikipedia.org/wiki/Mathematical_coincidence

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

that's (the/a) equation for finding an approximation of e, a constant in mathematics.


Hang on, I'd reword that as "that equation happens to result in an approximation of e".

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).

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

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.

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

Not enough people have downvote powers and too many people have upvote powers.

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.

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

I was surprised about how relevant the google results were actually! http://en.wikipedia.org/wiki/Mathematical_coincidence

(pi^4+pi^5)^(1/6) is a coincidental approximation to the mathematical constant e

Yes, you're missing something. The "2.xxx..." is remarkably close to the mathematical constant e. See my comment[0] for more.

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

Ah. Reminds me of http://xkcd.com/217 :)

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

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

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?

This is also highly dependent on how you define a day. A sidereal day is < 24 hours.

More interestingly, 10! seconds is exactly 6 weeks.

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

i th root of i

Not what you would expect.

Which is the same as i^(i^(i^i))).

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