
What's up with the number 163? - ColinWright
https://plus.google.com/101584889282878921052/posts/7u73y5FzEZY?pid=5963315119367216690&oid=101584889282878921052
======
nilkn
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.

~~~
lutusp
> 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](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.

~~~
nitrogen
_For example, this integral is equal to Pi:

[http://i.imgur.com/tJS1ZaA.png](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?

~~~
lutusp
Yes -- well, 1/2 the area of a unit circle, which is why I multiply by 2. I
used Sage ([http://www.sagemath.org/](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](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/](http://www.arachnoid.com/calculus/)

~~~
kleim
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](https://www.google.com/search?q=y+%3D+sqrt%281-x^2%29)

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

[http://www.wolframalpha.com/input/?i=2+integrate%28sqrt%281-...](http://www.wolframalpha.com/input/?i=2+integrate%28sqrt%281-x%5E2%29%2Cx%2C-1%2C1%29)

------
mproud
In Major League Baseball ([http://mlb.com](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_...](https://en.wikipedia.org/wiki/List_of_Major_League_Baseball_tie-
breakers)) 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.

~~~
KC8ZKF
In 1997 interleague play screwed up that nice round robin, among other things.

~~~
mproud
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.

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ccvannorman
Math continues to fascinate me. Here's another one I love:
[http://xkcd.com/217/](http://xkcd.com/217/)

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

~~~
legohead
[https://www.youtube.com/user/numberphile](https://www.youtube.com/user/numberphile)

~~~
chime
Love that channel. Add to that list:
[https://www.youtube.com/user/computerphile](https://www.youtube.com/user/computerphile)

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xamolxix
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.

~~~
draugadrotten
> 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/](http://www.imdb.com/title/tt0138704/)

~~~
lutusp
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. :)

------
impendia
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.

------
gus_massa
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?

~~~
wbl
It's the coefficient of the linear term in the q-expansion of the j-invariant.
The j-invariant of i _sqrt(163) is an integer exactly, and e^{pi_ sqrt(163)}
is the first term.

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

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lucb1e
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?

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

[https://plus.google.com/101584889282878921052/posts/7enyPxZW...](https://plus.google.com/101584889282878921052/posts/7enyPxZW3RB)

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

~~~
mikegioia
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](http://imgur.com/sFDCt7K)

Here's the comment:
[http://pastie.org/private/v5jg4cs014bojf1synjgq](http://pastie.org/private/v5jg4cs014bojf1synjgq)

~~~
soneca
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.

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ColinWright
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...](https://lh4.googleusercontent.com/-h0MYU3EVETo/UsHzgY2ZXjI/AAAAAAAAVsQ/j4isMZBaR84/w1106-h654-no/163-f.png)

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.

~~~
scythe
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...

------
kyllo
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](http://en.wikipedia.org/wiki/NetEase)

~~~
yzzxy
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).

See also: Japanese number puns

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

~~~
jackvalentine
I can still remember the McDonalds home delivery number in Shanghai and I
never called it, purely from hearing it on TV a few times two years ago:

4008517517 si ling ling ba wu yao chi wu yao chi "我要吃我要吃“

~~~
madcaptenor
McDonald's delivers in Shanghai?

~~~
kyllo
On scooters / mopeds I believe.

------
0-o
This is like the Ancient Aliens series for numerology. For episode 2 I suggest
the number 2. As a premeditation I suggest we all start to cogitate on the
preternatural property of the number 2: it being a "magic number".

------
nobrains
There is a question in QuizUp which says "Where does 163.com take you?"

------
e3pi
Another cautionary tail:

(pi^5+pi^4)^(1/(3+2+1)) = 2.7182818...

------
danbmil99
Math: fuck yeah!

------
ivanche
So you can always ask your girlfriend "Do you want 163 * (pi - e)" :)

