What's up with the number 163? | Hacker News Reader

What's up with the number 163?

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110 pointsColinWright58 comments

I'll be honest -- I don't find this sort of thing terribly interesting. I feel when reading this as if I'm being pressured to ooh and ahh, but to me the magic is just not there.

It also strikes me as backwards that people are referring to these incredibly deep algebraic connections as explanations. They're not explanations -- they're just connections. I don't need any of that stuff to prove these identities. I just need a pocket calculator. The algebraic connections might be useful for discovering these identities in the first place, but are not necessary for proving anything here. For any given arithmetic identity I'm sure you can find some remarkably obtuse algebraic connection which illustrates the same identity. These connections are interesting only insofar as the other stuff is interesting. The Monster group is interesting, but this identity itself is not particularly amazing to me.

Finally, is .999 really "extremely" close to an integer? In engineering, maybe. But in pure mathematics that seems "extremely" far from an integer.

> I don't need any of that stuff to prove these identities. I just need a pocket calculator.

But numerical results don't establish identities, and identities don't depend on numerical results. After all, with only numerical results for guidance, all sorts of illusory outcomes appear to mean more (or less) than they do. For example, this integral is equal to Pi:

http://i.imgur.com/tJS1ZaA.png

If one approximates the above integral by summing a bunch of numerical results on the interval between -1 and 1, one can approximate Pi, but the above integral equals Pi -- it's an identity. It has all sorts of advantages over a numerical result, not least of which is the fact that one cannot express Pi using a finite number of decimal places.

The article's examples weren't really identities, they were just coincidences (for the most part).

> Finally, is .999 really "extremely" close to an integer?

This is why mathematics has it all over words for expressing certain ideas.

For example, this integral is equal to Pi:

http://i.imgur.com/tJS1ZaA.png

It's been years since I've done any calculus. Do I understand that integral correctly as finding the area of a circle of radius 1, which is π?

Yes -- well, 1/2 the area of a unit circle, which is why I multiply by 2. I used Sage (http://www.sagemath.org/) to create these quick results:

    f(x) = sqrt(1-x^2)
    2 * integrate(f(x),x,-1,1)

Sage replies: pi (Sage is smart enough to knows this result is equal to pi, so it doesn't provide a numerical approximation, but simply prints "pi").

    plot(f(x),x,-1,1,gridlines=True,aspect_ratio=1)

Sage replies: http://i.imgur.com/GZdnT8z.png

The function by itself produces a line equal to 1/2 the distance across a unit circle at the coordinate provided by x. Therefore a definite integral of the function on the interval -1 to 1 should produce an area equal to 1/2 that of a unit circle, or pi/2.

My Calculus tutorial: http://www.arachnoid.com/calculus/

Or you can do it quick and dirty with a simple Google search: https://www.google.com/search?q=y+%3D+sqrt%281-x^2%29

A nice, convenient way to get the graph. For the integral:

http://www.wolframalpha.com/input/?i=2+integrate%28sqrt%281-...

tl;dr yes. I just spent the past few minutes trying to reverse engineer it. I'm no expert, but here's how I figure things. For the left side of the equation:

The formula for a circle's area is a = PI r^2. For a unit circle, the area is simply equal to PI.

Now for the right side of the equation:

r^2 = x^2 + y^2 is the Pythagorean Theorem and also the equation of a circle. If we solve for y, we have y = sqrt(r^2 - x^2). In other words, the height of any circle (centered at the origin) is sqrt(r^2 - x^2) given any particular place along the x-axis. This is the same expression which is integrated.

For a unit circle, we assume r = 1 and ignore everything outside of -1 and 1 on the x-axis. This is why the integral is bound between -1 and 1. So now we have integrate(-1, 1){ sqrt(1^2 - x^2) }.

By integrating we get the area of the top half of the unit circle. But we also need the bottom half. So we just double everything and that's what the extra 2 is for. We now have the area of a unit circle on the right side too.

QED

All good. :)

These identities are numerical results. Your integral example is not really the same sort of identity.

For that reason, I agree with you that "identity" is a bit of a misnomer here.

> These identities are numerical results.

You missed my point -- the examples in the linked article aren't identities, not as mathematicians use that word.

http://en.wikipedia.org/wiki/Identity_(mathematics)

I responded to that part of your post as well. I am fully aware of what "identity" normally means in mathematics.

I didn't feel the need to make a big deal out of the word in my original post since it's somewhat tangential and I thought it'd be clear what I was referring to (since the author of the post uses the same terminology).

Your point that true identities can't be proven with numerical calculations is true but not relevant, because as we're both aware these identities are not the pure sort of identity you had in mind when you wrote that post.

On the other hand I find these results fascinating. They are like a microcosm of experimental science. Numerical coincidences like these can be found by any high school student with a calculator, but only some deep theory can tell us whether they really are coincidences, in the statistical sense, or whether they are a manifestation of some precise underlying structure.

The Ulam spiral is one of my favourite examples of this kind of thing. http://en.wikipedia.org/wiki/Ulam_spiral

Most interesting things about number theory are only proven or even approached by these deep methods. I sometimes say that the history of mathematics is mostly about finding ways to not do number theory.

I think maybe you're misunderstanding me. The deep methods are interesting, but these numerical results are just 'meh' to me. These results aren't proven at all by deep methods.

These numerical comparisons are very, very different results than something like, say, the prime number theorem, which is basically the opposite. It can be conjectured with a pocket calculator, but requires deep methods to prove. These results can be proven with a pocket calculator, but can be discovered via deep methods (but also via trial and error).

In Major League Baseball (http://mlb.com), the season lasts 162 games, from late March/early April to the very end of September. Game 163 is played, only if necessary, as a tiebreaker game (https://en.wikipedia.org/wiki/List_of_Major_League_Baseball_...) that usually determines which team is heading to the postseason and which team is “going home.”

Why 162? In 1961, the American League expanded from 8 teams to 10 teams, so instead of playing 154 games against 7 different opponents (22 each), a team now played each other 18 times. (This also works well because 18 is a factor of 3, and baseball often plays its games in a series of 3, held at the same ballpark.) The next year the National League followed suit.

If there are any Game 163’s this season they’ll happen Monday.

In 1997 interleague play screwed up that nice round robin, among other things.

Yeah, interleague play, as certainly now with the odd number of teams in each league has prevented a good, consistent number of games between teams — though more teams and divisions didn’t help either.

Math continues to fascinate me. Here's another one I love: http://xkcd.com/217/

I've spent hours in front of Google trying to find other random symmetries.. :]

https://www.youtube.com/user/numberphile

Love that channel. Add to that list: https://www.youtube.com/user/computerphile

I've just spent all afternoon watching numberphile videos.

Since numbers are infinite and the ways we can manipulate them are infinite (or at least large) doesn't it mean that unlikely or strange things happen all the time?

I would be more surprised if they wouldn't happen.

Yes. This is sometimes called The Strong Law of Small Numbers, see e.g. http://mathworld.wolfram.com/StrongLawofSmallNumbers.html and https://www.maa.org/sites/default/files/pdf/upload_library/2...

Speaking analogously, "If we gave everyone one in the world a device that randomly shot paint at the wall, I would be more surprised if no one ended up with an interesting pattern."

I agree, but whatever that pattern (or patterns) turned out to be is still an interesting question.

> Since numbers are infinite and the ways we can manipulate them are infinite (or at least large) doesn't it mean that unlikely or strange things happen all the time?

They do. If you haven't seen it already I can recommend a movie about this topic. It can be enojyed for other reasons as well.

Pi: "A paranoid mathematician searches for a key number that will unlock the universal patterns found in nature." http://www.imdb.com/title/tt0138704/

I like the movie, it deserves its fame, but there are some errors in the script that could have been easily avoided. For example, at one point a character says something to the effect of, "Surely they've printed out every 216-digit number?" No, not likely. :)

+1 for the movie Pi recommendation---it's proper good stuff.

It means that it would be weird for there to not be interesting patterns all over the place, as you suggest. This is, however, subtly different from there being such patterns. In particular, the first one is stated independent from the effort it would require to find them, while the latter includes it.

In other words, maybe the right want to understand this is via intuitionistic logic where not (not A) is not convertible to A. So, it's certainly not the case that there are no interesting patterns... but you must show me one before I believe that they actually do exist.

Here's another directly related phenomenon.

Look at the function f(n) = n^2 + n + 41. Now start plugging in nonnegative integers. You get 41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, .....

All primes?! Well, not quite, plug in n = 41. But the fact that you get primes for so long is quite directly (but in a rather complicated way) explained by what ColinWright describes.

From the article:

> e^{pi * sqrt(163)} ~= (5280 x 5280 x 5280) + 744

> e^{pi * sqrt(67)} ~= (640320 x 640320 x 640320) + 744

I really have to ask: What's up with 744?

It's the coefficient of the linear term in the q-expansion of the j-invariant. The j-invariant of isqrt(163) is an integer exactly, and e^{pisqrt(163)} is the first term.

Now I want a t-shirt where the front reads "What's up with 744?" and the back is your response.

I use your answer to guide my google searches. The easier explanation I found is in the "details" section of: http://en.wikipedia.org/wiki/Heegner_number#Almost_integers_...

Right. Well, obviously.

Off to google...

You have these equations reversed.

Yes. Too late to edit :(.

Better yet, what's up with Google+? I get kicked back to Hacker News when I try to close the photo overlay. How to view the original page?

Yeah, that's a rather obnoxious behavior. Perhaps someone can change the link to the following?

https://plus.google.com/101584889282878921052/posts/7enyPxZW...

I always open in a new tab, and then closing the photo takes me to here:

https://plus.google.com/101584889282878921052/posts/7u73y5Fz...

Is that what you intended?

Am I the only one directed to a blue tiled dome post? Weird, i really don't understand.

From a comment by peterjmag near the bottom: https://news.ycombinator.com/item?id=8379877

The direct link is https://plus.google.com/101584889282878921052/posts/7enyPxZW...

I don't have G+, so I'm guessing. IIUC the submitted link goes to a slideshow over a unrelated G+ post. Maybe C.W. saw the thumbnail of the slideshow from the other post, clicked "see/expand" and got the slideshow in the overlay.

The unrelated post with the blue dome is 7u73y5FzEZY

The actual post with the 163 equations is 7enyPxZW3RB

So if we change the URL to this, we can get the best link: https://plus.google.com/101584889282878921052/posts/7enyPxZW...

I saw the same blue tile post on my phone. I think some wires got crossed somewhere. The link gets redirected correctly on my desktop, but google says it encountered an error.

Here's the image: http://imgur.com/sFDCt7K

Here's the comment: http://pastie.org/private/v5jg4cs014bojf1synjgq

Thanks. A comment below about the problem with closing a photo and directed back to hn also directs correctly.

A post technically explaining what is happening would be worth my upvote.

Link is just broken on mobile. I was also very, very confused

> Link is just broken on mobile.

It's an example of narrowcasting -- it requires very specific browser features to function as intended. And even with such a browser, IMHO it's not the most efficient way to present its content. Whatever happened to old-fashioned static HTML?

For those struggling with the link and with Google+ (why is it so crap?) here is the image:

https://lh4.googleusercontent.com/-h0MYU3EVETo/UsHzgY2ZXjI/A...

The text says:

What's up with 163?

The number 163 is involved in a number of mathematical formulae that, for some reason, produce answers that are extremely close to integers.

The first identity shows that 163 x (π - e) is extremely close to the integer 69. Here, π and e are famous constants, approximately equal to 3.141592653589793... and 2.718281828459045..., respectively.

The second identity shows that 163/ln(163) is even closer still to the integer 32. Here, “ln” denotes the natural logarithm; that is, the logarithm to base e.

The number in the third identity is almost freakishly close to the integer (640320 x 640320 x 640320) + 744. Actually this is not a coincidence, but the reason that it happens is very deep and has something to do with the Monster simple group. The Monster is perhaps best understood in terms of the symmetries of a 196884-dimensional algebraic object. Yes, that's as bad as it sounds.

Fortunately, there is a way to understand what is so remarkable about the number 163 in terms of unique factorization. What does that mean? Well, an important and very useful property of the integers is that every integer greater than 1 can be written as a product of primes in an essentially unique way. For example, the integer 21 can be written as 3 x 7, or as 7 x 3; we don't consider these factorizations to be essentially different because the only difference between them is that the factors appear in different orders.

This unique factorization property is still true if we consider negative integers and think of negative prime numbers as prime. We could then factorize -21 as (-3) x 7, 3 x (-7), 7 x (-3) or (-7) x 3. We consider these factorizations to be essentially the same, because they only differ in (a) the order of the factors and (b) multiplication of the factors by the (multiplicatively) invertible integers +1 and -1.

Unique factorization is a rare property for a number system to have.

What does this have to do with 163? Well, the number 163 is the largest of the nine Heegner numbers; the complete list of these is 1, 2, 3, 7, 11, 19, 43, 67, 163. If d is a Heegner number, then we can enlarge the rational number system Q by adding a square root of -d. This enhanced version of the rational numbers comes equipped with an enhanced system of integers, called the ring of integers. This ring of integers can be constructed from the usual integers by adjoining the square root of -d, sqrt(-d), unless -d-1 is a multiple of 4, in which case we adjoin the number (1 + sqrt(-d))/2 instead.

Heegner proved in 1952 that the ring of integers constructed above has the unique factorization property if and only if d is a Heegner number. The other large Heegner numbers also give identities like the one in the picture. For example, if we replace 163 by 67 in the third identity, the result is within 0.0000013 of the integer (5280 x 5280 x 5280) + 744.

If you replace 163 by 67 in the second identity, you get 67 / log(67) ≈ 15.93. I had heard the explanation for e^(pi sqrt(x)) in terms of the j-function, but I don't know where this x / log(x) thing is coming from...

First thing I thought of was 163.com aka NetEase, one of the most popular Chinese web portals (basically like the Yahoo! of China): http://en.wikipedia.org/wiki/NetEase

Most Chinese sites use arabic numerals as their domain names - Hanzi domains are pretty new, and there's some cool overlap between number pronunciation and that of normal words in Mandarin (Mandarin is pretty much just monosyllable words, so there's a lot of overloaded meaning).