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:
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.
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.
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.
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.
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.
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/
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.
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.
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.
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
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.
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).
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".
[+] [-] nilkn|11 years ago|reply
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|11 years ago|reply
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.
[+] [-] Ended|11 years ago|reply
The Ulam spiral is one of my favourite examples of this kind of thing. http://en.wikipedia.org/wiki/Ulam_spiral
[+] [-] lmm|11 years ago|reply
[+] [-] mproud|11 years ago|reply
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|11 years ago|reply
[+] [-] ccvannorman|11 years ago|reply
I've spent hours in front of Google trying to find other random symmetries.. :]
[+] [-] legohead|11 years ago|reply
[+] [-] xamolxix|11 years ago|reply
I would be more surprised if they wouldn't happen.
[+] [-] jks|11 years ago|reply
[+] [-] Totient|11 years ago|reply
I agree, but whatever that pattern (or patterns) turned out to be is still an interesting question.
[+] [-] draugadrotten|11 years ago|reply
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/
[+] [-] tel|11 years ago|reply
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.
[+] [-] impendia|11 years ago|reply
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|11 years ago|reply
> 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|11 years ago|reply
[+] [-] jessaustin|11 years ago|reply
[+] [-] lucb1e|11 years ago|reply
[+] [-] peterjmag|11 years ago|reply
https://plus.google.com/101584889282878921052/posts/7enyPxZW...
[+] [-] ColinWright|11 years ago|reply
https://plus.google.com/101584889282878921052/posts/7u73y5Fz...
Is that what you intended?
[+] [-] soneca|11 years ago|reply
[+] [-] gus_massa|11 years ago|reply
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...
[+] [-] mikegioia|11 years ago|reply
Here's the image: http://imgur.com/sFDCt7K
Here's the comment: http://pastie.org/private/v5jg4cs014bojf1synjgq
[+] [-] forrestthewoods|11 years ago|reply
[+] [-] ColinWright|11 years ago|reply
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.
[+] [-] scythe|11 years ago|reply
[+] [-] kyllo|11 years ago|reply
[+] [-] yzzxy|11 years ago|reply
See also: Japanese number puns
http://en.wikipedia.org/wiki/Japanese_wordplay
[+] [-] 0-o|11 years ago|reply
[+] [-] nobrains|11 years ago|reply
[+] [-] e3pi|11 years ago|reply
(pi^5+pi^4)^(1/(3+2+1)) = 2.7182818...
[+] [-] danbmil99|11 years ago|reply
[+] [-] ivanche|11 years ago|reply