Talk:Mathematical coincidence

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1 Name of article
2 From a concerned mathematician
3 Two levels of "coincidence"
4 The title of this page
5 these are not coincidences
6 Not-Coincidences Removed
- 6.1 reversion of mass delete
7 Scope of page
- 7.1 Coincidences or approximations
8 Approx vs. Simeq notation
9 Ranking
10 Other curiosities
11 sources
12 Defining coincidence
13 Error
14 Spelling of 40

[edit] Name of article

A few comments. Maybe mathematical coincidence might yet be a better page name. I'm not sure what a 'theoretical explanation' would be, in all cases: but certainly in the case of exp(π√163) there is a very good if hidden reason. Perhaps the pigeonhole principle could be invoked. After all, it is because of Dirichlet's theorem on diophantine approximation, based on it, that we know that good rational approximations exist. And so on, for other kinds of relations. By the way, I added the law of small numbers link, but that really needs disambiguation since the Poisson distribution meaning is a separate thing.

Charles Matthews 16:10, 21 Jun 2004 (UTC)

Hi Charles

you're probably right about the page name. I chose the name originally because I reckoned that there would be hundreds and hundreds. In the end, the best I could do wasn't very extensive! There must be more out there! (exactly the same thing happened with List of scientific howlers in literature)

The scientific howlers list is likely to grow, slowly but surely. Dpbsmith 22:58, 21 Jun 2004 (UTC)

Anyway, the exact definition of a mathematical coincidence is problematical. I spent some considerable time pondering the best definition, only to get myself tangled up in philosophy, and at one point defining everything to be a coincidence. Of course $\pi^2\simeq 10$ ! How could it possibly be otherwise? Nevertheless, there is definitely _something_ that all the examples have in common. Except maybe the exp(pi*sqrt(163)) one, which as you point out is a result of some algebraic number theory (all of which I forgot immediately after my finals). And possibly the continued fraction ones, although it _is_ coincidental that the continued fraction for pi has a large coefficient early on (isn't it?)

Best wishes

Robinh 20:51, 21 Jun 2004 (UTC)

[edit] From a concerned mathematician

I am a mathematician who studies (among other things) topological (Nielsen) coincidence theory. The word "coincidence" has a widely accepted and recognized technical meaning: given two functions f and g, the Coincidence Set Coin(f,g) is the set of points x such that f(x) = g(x).

This is important as it is (perhaps) the most natural generalization of fixed point (mathematics) theory (where g is taken to be the identity map) and is the original setting for many results (e.g. the Lefschetz fixed-point theorem) which are now commonly misidentified as being fixed point results. (Lefschetz proved first the coincidence version of his result, and noted the fixed point version as a specific case.)

Coincidence theory also has an important reduction when the function g is taken to be a constant map- this setting is called root theory (finding points x with f(x) = c for some constant c).

As a mathematician, I feel like a wikipedia page on coincidences should refer to the above concept. The content now described as "mathematical coincidence" would perhaps be more accurately described as "mathematical curiosity".

My $0.02. Chris Staecker, UCLA Math Dept

Well, we can have a page coincidence set, or coincidence (mathematics), any time you want. Charles Matthews 19:30, 26 September 2005 (UTC)

If you ever decide to make those pages, I trust no one would object to the placement of a link to it at the top of the page instructing people on ambiguity of the title.Wrath0fb0b 14:26, 16 March 2006 (UTC)

[edit] Two levels of "coincidence"

It seems to me that there's an important distinction between "coincidences" that arise out of a general theory, like the one about exp(pi sqrt 163), and ones for which no explanation of any kind is known. To take a few examples from the page as of 2005-04-10:

e^pi ~= pi^e is true because pi is close to e and x / log x is stationary at e. Not much of a coincidence, really; the same would be just as true if 3 or sqrt(10) were used in place of pi.
pi ~= 355/113 is a "real" coincidence, so far as I know: that is, no reason is known why pi should have so large a coefficient so early in its continued fraction.
sqrt(2) ~= 17/12 is not much of a coincidence; there's a general theorem that says that roots of quadratic equations always have rational approximations that are at least about that good.
exp(pi) ~= pi+20 seems to be a "real" coincidence.
1 mile / 1 km ~= phi is an absolute coincidence, if anything is. (Incidentally, it might be worth mentioning the Zeckendorff representation in connection with this...)

Is it worth making this distinction on the page? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)

You make an excellent point. One might as well claim that "pi ~= 3.14" is a coincidence. How about pi^4 + pi^5 ~= e^6? MrHumperdink 16:24, 7 May 2006 (UTC)

Cooool... I'm adding this to the article.--Army1987 18:59, 7 May 2006 (UTC)

It is cool, but many of these are not. pi ~= 22/7 is not a coincidence, it is an approximation. I'm going to be bold and go through the list, removing what do not seem to be coincidences. Obviously my point of view should be rechecked by other editors; I'll list what I take out here. MrHumperdink 00:31, 8 May 2006 (UTC)

[edit] The title of this page

How about "numerical coincidences"? Gareth McCaughan 13:48, 2005 Apr 10 (UTC)

[edit] these are not coincidences

None of these can be classified as coincidences, except maybe

$e^\pi\simeq\pi^e$ . -Hmib 02:48, 20 Jun 2005 (UTC)

which is explained by the fact that e and pi are both close to 3. --MarSch 13:39, 28 October 2005 (UTC)

[edit] Not-Coincidences Removed

I have removed the following items from the list:

- $\sqrt{2 \pi} \simeq 5/2$ to about 0.1% (one part in a thousand).
- $\pi\simeq 22/7$ ; correct to about 0.04%; $\pi\simeq 355/113$ , correct to six places or 0.000008%.
- $\pi^3\simeq31$ ; correct to about 0.02%.
- $\pi^2\simeq10$ ; correct to about 1.3%. This coincidence was used in the design of slide rules, where the "folded" scales are folded on $π$ rather than $\sqrt{10}$ , because it is a more useful number and has the effect of folding the scales in about the same place; $\pi^2\simeq 227/23$ , correct to 0.0004% (note 2, 227, and 23 are Chen primes).
- $\pi^4\simeq 2143/22$ , accurate to about one part in $1010$ ; due to Srinivasa Ramanujan, who might have noticed that the continued fraction representation for $π 4$ begins $[97; 2,2,3,1,16539,1,1,\ldots]$ . See also: Pi culture.
- $\pi^5\simeq306$ ; correct to about 0.006%.

Up to here, all of these are simply approximations. So pi is close to 3 and slightly greater... obviously if you square it you get close to 10. Similarly, one might as well call it a "coincidence" that pi ~= 314/100, as to consider 22/7 and and the other fractions.

Continued:

- ; leading to Donald Knuth's observation that, to within about 5%, log₂(x) = log(x) + log₁₀(x).
  - This one I'm unsure about. I took it out because it seems too complex or contrived to qualify.
- π seconds is a nanocentury (ie 10 ^{- 7} years); correct to within about 0.5%
  - Yes, or 3, or 3.1, or... basically, nothing special about pi here.
- one attoparsec per microfortnight approximately equals 1 inch per second (the actual figure is about 1.0043 inch per second).
  - "Microfortnight"? No.
- a cubic attoparsec (a cube where each edge is one attoparsec) is within 1% of a fluid ounce.
  - Again, what's special about an attoparsec, or a microfortnight, or whatever?
- ; accurate to 9 decimal places (due to Ramanujan).
  - What an astounding coincidence.
- N_A ≈ 2⁷⁹, where N is Avogadro's number; correct to about 0.4%. This means that a yobibyte is slightly more than two moles of bytes.
- The speed of light is about one foot per nanosecond (accurate to 2%) or 3×10⁸ m/s (accurate to about 0.1%)

Well, that's it for now. I tried to look for similarities which were neither too simple (e + 0.4 is close to pi! Stop the presses!) or too complex (see Ramanujan's approximation of pi). I hope this helps... MrHumperdink 01:08, 8 May 2006 (UTC)

[edit] reversion of mass delete

Hi. You removed 13 points. I disagree with 10 of these deletions (which I consider to be valid entries), and am unsure about the other 3. Perhaps a good way forward would be to delete entries one at a time, and then we discuss each one on its merits.

best wishes, Robinh 07:23, 8 May 2006 (UTC)

Please, tell me why you consider these to be coincidences, rather than simply reverting my edit in its entirety. I have discussed each one's merits. It is your place to respond now. Approximations are not "coincidences". They can be explained. That pi^4 + pi^5 ~= e^6 is pretty hard to explain, and is relatively simple. And so on. MrHumperdink 14:50, 8 May 2006 (UTC)

I am for removal of approximations - they are not coincidences. What so special about pi = 22/7? Is 22 somewhat special, or 7? Better mention some (not really coincidences, but amazing properties of numbers) other things, like friendly numbers: 220 equals sum of all (proper) divisors of 284 (1+2+4+71+142) , while 284 equals the sum of all (proper) divisors of 220 (1+2+4+5+10+11+20+22+44+55+110). --Alex ^talk 15:55, 8 May 2006 (UTC)

That makes two of us, then. Robinh, please state your reasons for reverting my edit. I won't start an edit war - no worries there :) - but I would like to continue the discussion. I'll leave you a message. MrHumperdink 02:20, 9 May 2006 (UTC)

The somewhat special thing is that pi has such a large term as 292 very early in its continued fraction, and that the first powers thereof all have big numbers quite early in their CFs, in the case of π^3 and π^5 as soon as the first denominator. --Army1987 19:58, 10 May 2006 (UTC)

[edit] Scope of page

This discussion and the page itself shows there is some uncertainty about the scope of this page. I can't really see there's any difference between a co-incidence and an approximation, except the former must at least seem slightly surprising in its closeness. The lack of a theoretical explanation is demanded in the introduction, yet we have the $e^{\pi\sqrt{n}}$ co-incidences included (possibly my favourite example). I suggest the lack of a theoretical explanation comment should be dropped, as its only effect would be to remove the most interesting examples.

Also, I would like to include "co-incidental" equalities, such as:

$\sum_{n=1}^{24} n^2 = 70^2\;$

a remarkable, unique and highly significant "co-incidence".

Elroch 22:06, 8 May 2006 (UTC)

Exactly my point. $e^{\pi\sqrt{n}}$ clearly should be removed (log(n+14) also passes near integers especially when n=163. So?). The mile being phi kilometers is quite a coincedence. The number pi as irrational number being well approximated by a member of "everywhere dense" subset of rational numbers (for example 22/7) is not. I would like to ask Robinh to remove all examples of "theoretically obvious" approximations. PLEASE. --Alex ^talk 05:09, 9 May 2006 (UTC)

[edit] Coincidences or approximations

Hello everyone.

sorry to revert a good-faith edit. (No problem. MrHumperdink 00:44, 10 May 2006 (UTC)) Well, first of all why not split the page into mathematical coincidence and mathematical approximation? This would have the advantage that $\pi^2\simeq 10$ (which is certainly notable from a slide rule perspective) would have a place in the encyclopedia. Would that be a good idea?

(Let's discuss the others when we've come to a conclusion on the pi ones)

On the other hand, I do feel that $\pi\simeq 22/7$ is a coincidence as well as an approximation. The interesting fact is that the second quotient in the continued fraction representation of pi (15) is large. There seems to be no explanation for this (interesting, notable) fact than "the CF expansion is what it is".

$\pi\approx 22/7$ is an approximation proven by Archimedes through calculating perimeter of a circumscribing 96-sided proper polygon. No real coincidences here. --Alex ^talk 15:40, 9 May 2006 (UTC)

Also, taking $\pi^4\simeq 2143/22$ and the other longer identity for pi. Now, Ramanujan himself considered these to be notable coincidences (although I must confess that I can't put my finger on an original source). Ramanujan was a genius and if he thought that a near-equality was notable enough to write down, then it's certainly notable enough for wikipedia.

best wishes, Robinh 07:21, 9 May 2006 (UTC)

1. On "Mathematical approximation" article. Not very good idea, at least with this name. There exist Approximation, Diophantine approximation, Thue–Siegel–Roth theorem to name a few on the topic.

2. While on the subject. I wonder why are you using symbol $\simeq$ instead of $\approx$ ?

3. $\pi^2\approx 10$ - is a coincidence, if you consider that an area of a unit circle has nothing to do with the number of fingers. But 22/7 is still an approximation and nothing more. 22 and 7 - some regular numbers, not representing anything else. I don't see significant coincidence: nothing cool about it.

4. We should take each approximate identity one-by-one and discuss it (with arguments, of course). Some may definitely survive, the rest should be removed. I agree that massive deletes are not a way to go, but some clenup is needed.

5. We can also take another approach, let participants rank "coincidences" by a level of "amusement" and absense of theoretical explanation. And lets kill the least "amusing" ones - those which are kind of expected and uninteresting. Surely, Ramanujan's long near identity for pi would be high on my list.

--Alex ^talk 13:58, 9 May 2006 (UTC)

How about few simple ones:

$\frac {16} {64} = \frac {1\not6} {\not6 4} = \frac {1} {4}$ , $\frac {26} {65} = \frac {2\not6} {\not6 5} = \frac {2} {5}$ , $\frac {19} {95} = \frac {1\not9} {\not9 5} = \frac {1} {5}$ :)--Alex ^talk 14:55, 9 May 2006 (UTC)

First: clever. Nice fractions there... :P I can agree about $\pi^2\approx 10$ now that I understand its truly coincidental quality. I still really disagree with 22/7 and even 2143/22. As I understand it, a "coincidence" is when two independent concepts (or events or whatever) suddenly meet. As Alex said, 22 and 7 are not "special" numbers. It's not enough to be unexpected (the birthday paradox, for instance, is "unexpected" but certainly not a "coincidence" by any meaning of the word); there must be two separate ideas put together. For instance, $\pi^4 + \pi^5\approx e^6$ is coincidental because it links two "special" and unconnected numbers unexpectedly. I really don't consider Ramanujan's approximation of pi. 63, 25 and 15sqrt(5)? How is it a coincidental expression? etc etc. MrHumperdink 00:44, 10 May 2006 (UTC)

[edit] Approx vs. Simeq notation

First things first. Symbol \Approx ( $\approx$ ) means approximately equal and should be used when dealing with numbers (for example $\pi \approx 3$ ). Symbol \Simeq ( $\simeq$ ) on the other hand means similar or equal, which is definitely not the case with fixed values like numbers, it is usually used with functions (for example $\sin x \simeq x$ ). I believe I am clear on this: all number relationships should use $\approx$ instead of $\simeq$ . When we will get to assymptotic approximations/coincedences for functions then we will use $\simeq$ . Ok? --Alex ^talk 03:09, 10 May 2006 (UTC)

no, not okay. $\approx$ means approximately equal and there is no reason why it couldn't be used for functions. $\simeq$ means homotopic.--MarSch 15:02, 11 May 2006 (UTC)

You better search Wikipedia for misuse of $\simeq$ then. Because I found $\sin x \simeq x$ in one of the articles. --Alex ^talk 17:13, 11 May 2006 (UTC)

[edit] Ranking

Lets rank article examples to determine least impressive and highly anticipated with the goal to remove them. Here are my choices. I consider highly non-coincidental (candidates for deletion) the following examples:

$\pi^3 \approx 31$ . What did you expect if $\pi^2 \approx 10$ ? 43? 55? Of course, it's almost 31.
$\pi^5 \approx 306$ . My six-years old knows that if $\pi^2 \approx 10$ and $\pi^3 \approx 31$ then $\pi^5 \approx 306$ . How about

$\pi^7 \approx 3020$ with precision of 0.01%? You can continue to infinity, just multiply by $\approx 10$ .

$\pi \approx 22/7$ . Every schoolboy in ancient Egypt knew this one - it kind of lost its element of surprise (if it ever had any).

Removal of these non-coincidental examples would be a good start. --Alex ^talk 03:29, 10 May 2006 (UTC)

pi^7 = 3020.2932.. is not very impressive. pi^3 = 31.006... definitely is. pi^5 = 306.019... probably is. --Army1987 18:36, 11 May 2006 (UTC)

pi^7 is no more or less impressive than pi^2 = 9.8696... and yet we include "pi^2 ~= 10".--MrHumperdink 18:07, 12 May 2006 (UTC)

I agree that, from a purely mathematical POV, pi^2 = ~10 is not very impressive. But I guess it is here because it's one of the approximate identities which have a practical usefulness. (However, I have no opinion on wheter to remove it from here.) --Army1987 13:13, 13 May 2006 (UTC)

Hi.

I think you're right about the pi powers. Trying the same thing with exp(n) gives similarly (un)surprising results.

But I'm unsure about $\pi\approx 22/7$ . The surprising thing about this is not the 22 nor the 7 per se, but the notable and interesting fact that in this close approximation, the "7" is small, due to the large quotient in pi's continued fraction expansion (15) occurring unusually early. OTOH, trying it with "random" nonquadratic irrationals gives a similar story. For example, $π + e$ (e itself is a bad idea as its CF representation is semi-regular) comes out with 41/7, which is about as good an approximation as 22/7 for pi.

Robinh 07:58, 10 May 2006 (UTC)

I'll give you this: $\pi\approx 22/7$ is somewhat of a coincidence, but it is not very impressive if it requires a paragraph of explanations. I change my position from "definitely remove" to "borderline boring". You can keep it, but don't stop looking for more interesting things. --Alex ^talk 15:05, 10 May 2006 (UTC)

It doesn't matter if the approximation is "a notable and interesting fact" but rather if it is coincidental. Ditch the approximations. I understand that the large denominators are unusual, but you have yet to show how they are part of an actual coincidence. --MrHumperdink 22:47, 10 May 2006 (UTC)

Robinh, the problem with approximations is that there is no feeling of uniqueness. You take some irrational number with large quotient and approximate it with the small denominator fraction claiming that it's close approximation. How rare is that? You mentioned

π + e

has as good approximation as

π

, so I assume all irrational numbers which differ from

π

somewhere beyound 100th digit have the same property. Nothing really special here. Nor beautiful. --Alex ^talk 04:08, 11 May 2006 (UTC)

Yes, that was my point: many irrationals that crop up in this line have similarly good approximations with similarly small denominators. Best, Robinh 07:03, 11 May 2006 (UTC)

If you want interesting fact about approximations, here it is: Golden ratio is the "most irrational" number in a sense that it has the slowest convergence of rationals with increasing denominators approximating it. No coincidence here, because

φ

is an infinite chain fraction of 1s

$1 + \frac{1} {1 + \frac{1} {1 + \frac{1} {...}}}$ --Alex ^talk 04:08, 11 May 2006 (UTC)

So... since you've agreed, Robinh, that the approximations have similar fractions, then there is nothing noteworthy about the powers of pi. I will wait a day, then remove them. --MrHumperdink 18:07, 12 May 2006 (UTC)

After a day, I removed the following five approximations:

$\sqrt{2 \pi} \approx 5/2$ to about 0.1% (one part in a thousand).
$\pi\approx 22/7$ ; correct to about 0.04%; $\pi\approx 355/113$ , correct to six places or 0.000008%.
$\pi^3\approx31$ ; correct to about 0.02%.
$\pi^4\approx 2143/22$ , accurate to about one part in $1010$ ; due to Srinivasa Ramanujan, who might have noticed that the continued fraction representation for $π 4$ begins $[97; 2,2,3,1,16539,1,1,\ldots]$ . See also: Pi culture.
$\pi^5\approx306$ ; correct to about 0.006%.

(The theory of continued fractions gives a systematic treatment of this type of coincidence; and also such coincidences as $2\times 12^2\approx 17^2$ (ie $\sqrt{2}\approx 17/12$ ). Curiously the continued fractions of the first few powers of

π

have big numbers (>50) quite early, in the case of

π 3

and

π 5

as soon as the first denominator.) --MrHumperdink 21:25, 13 May 2006 (UTC)

More on continued fractions. Periodic continued fractions (which are quadratic irrationals) explain and estimate the goodness of Ramanujan approximation $\pi\approx\frac{63}{25}\left(\frac{17+15\sqrt{5}}{7+15\sqrt{5}}\right)$ . Perhaps, in his time it was hard to discover things like that, but now well-studied periodic continued fractions allow to produce any number similar expressions. --Alex ^talk 06:32, 14 May 2006 (UTC)

So, I propose waiting a day again, in case Robinh has a comment. Then we can remove that one. --MrHumperdink 17:39, 14 May 2006 (UTC)

That one looks very artificial, I agree with removing it. Anyway: 1) I don't understand why the ones above were removed but pi^2 = 227/23 remains; 2) IMO that ones should be re-added, especially the one about pi^4, such a large term so early in its CF is unexpectable. --Army1987 19:42, 14 May 2006 (UTC)

Hi; thanks for waiting. You guys gonna love this. I've found the original reference in a paper of Ramanujan's of 1914. The expression is the twenty-fifth in a series of increasingly accurate approximations to pi. Ramanujan states, and I quote, "we cannot expect a high degree of approximation for small values of

n

". So I vote delete (and indeed I'm gonna delete it myself, on the grounds that noone else has ever expressed any support for keeping it :-)

Now, there is an interesting outcome too. The next section in the same paper states, and I quote directly, "Another curious approximation to pi is $\left(9^2+\frac{19^2}{22}\right)^{1/4}$ . This value was obtained empirically, and it has no connection with the preceding theory". This approximation is equivalent to

π 4 = 2143 / 22

, which I'll re-add to the article, with a reference. Best wishes, Robinh 07:22, 15 May 2006 (UTC)

That one can be stated another way: write 1234, swap the first two digits, swap the last two digits, divide by two-two, take the 2nd (square) root two times, and you get approximately pi. (That one was on the article pi a long time ago.) --Army1987 15:11, 15 May 2006 (UTC)

[edit] Other curiosities

Added section on amusing equalities. Not sure it belongs to this article, but it is fun reading. --Alex ^talk 15:36, 10 May 2006 (UTC)

This new section seems closest to what the article is getting at: unexpected equalities. I especially liked (1+9+6+8+3)^3=19683. That is a coincidence, I think. And, as you noted, they are actually interesting to hear.

Please, can we remove the Ramanujan one I keep mentioning? Or at least explain why it is coincidental? --MrHumperdink 22:47, 10 May 2006 (UTC)

[edit] sources

This article does not cite any sources. No coincidence, perceived or real, can be included without them. --MarSch 15:05, 11 May 2006 (UTC)

Do we really need them? Anybody can test them with a calculator. As WP:NOR says, it makes descriptive claims the accuracy of which is easily verifiable by any reasonable adult without specialist knowledge. --Army1987 18:39, 11 May 2006 (UTC)

But how do you verify that any of these coincidences are noteworthy, lacking "direct theoretical explanation"? Fredrik Johansson 20:30, 13 May 2006 (UTC)

You raise a good point - for us to rate these as interesting is POV and original research. We should probably take a leaf out of the book of inherently funny word, where the idea is to find "words considered to be inherently funny". That said, I don't really know where to look to find any authoritative source. As noted above, some people may consider certain similarities "coincidental", when they are not. (For instance, e^pi ~ pi^e, which is the case simply because e ~= pi.) I'll check Mathworld, see if they have an article there. --MrHumperdink 21:13, 13 May 2006 (UTC)

We are providing only examples of curious numeric facts. We don't pretend to list all "mathematical coincidences". From that point of view any correct fact found in valid sources that mark it as such is OK for inclusion (availability of primary source should address "original research" concern). This very discussion about merits of each individual example (in my mind) addresses NPOV concern. --Alex ^talk 06:40, 14 May 2006 (UTC)

[edit] Defining coincidence

I'm still rather unsure about the boundary between a coincidence which has an "explanation" and one which does not. dictionary.com defines coincidence as "The state or fact of occupying the same relative position or area in space", which seems more appropriate than the alternative definition about chance events. No mathematical coincidence is a chance event, since they are all determined precisely by the underlying axioms, with no uncertainty. I am sure that for any apparently "chance" co-incidence, someone could come up with an indirect argument why the approximation was as good as it is, rather than just doing the calculation and seeing that it is. Just because an approximation is "explained" by a large term in a continued fraction expansion is not a reason to exclude it, since good approximations will obviously tend to be associated with such continued fractions.

There should be some role for information theory here - a coincidence could be defined as an unusually compressed form for a good approximation. For example, a certain amount of information is needed for a fraction, and this amount of information can be compared to the error in an approximation. (It may be best to just compare the information content of the denominator to the accuracy) Elroch 11:36, 15 May 2006 (UTC)

Obviously all approximate equalities are determined and chance plays no role. But we could define a coincidence as an unexpected approximate equality. Yes, in the case of CFs with large terms, coincidences do have a reason, but an unexpected one. Nobody would expect the CF of pi^4 to have a five-digit term so early before calculating (or reading about) it. In the case of approximation of irrational number by a fraction, we could for example define a coincidence as when a term in the CF is much bigger than the denominator of the fraction obtained with the terms before it. For example, IIRC, pi = [3, 7, 15, 1, 292, 1...]; fifteen is more than twice as much as the denominator in 22/7, and two hundred and ninety-two is more than twice the denominator of 355/113. (Of course, this way many approximate equalities with integers could be deemed coincidences; but we could say that this shouldn't apply to integers, and approximate equalities with integers should be considered coincidences only when a number is within 0.02 of an integer.)

Anyway, I'm removing pi^e = e^pi. That's definitely not a coincidence, it happens with any two numbers enough close to e. For example, 2.5^3 = 3^2.5 is much more precise.

If nobody objects, I'm going to remove that one by Ramanujan with fractions and roots, and to re-add the ones about pi, pi^3, pi^4 and pi^5.--Army1987 14:52, 15 May 2006 (UTC)

I still don't see how they are "coincidental" rather than simply "unusual". Is there even a distinction to make? For instance, the famous

e i * π + 1 = 0

seems almost "coincidental" despite it having a logical proof. I don't know any more. I may leave this page up to you two... --MrHumperdink 23:05, 15 May 2006 (UTC)

[edit] Error

Towards the bottom of the page, it is stated that 71^3 = 357911, which is correct. However, this number is referred to as "consecutive primes", which is incorrect. —The preceding unsigned comment was added by mstemper (talk • contribs).

Good call. I suppose you could say "consecutive odd numbers", but it doesn't seem much of a coincidence. Why not list a factorization of 3579? or 35791113? It's somewhat remarkable that the given number was a perfect cube, but nothing to write home about. Staecker 01:43, 27 May 2006 (UTC)

I guess it meant three five seventy-nine eleven, but so, why are they "consecutive"? --Army1987 08:53, 27 May 2006 (UTC)

357911 also consists of three primes (which is amazing since the LHS contains a third power), when written in base 111 (three ones in a row!). Pick any couple of numbers and you can make up some "coincidence" between them if you think hard enough. Entries like this one are pointless. Fredrik Johansson 09:22, 27 May 2006 (UTC)

I've spotted two items which I believe are incorrect. The first one is the second entry on the page, "Exp[pi * sqrt(n)] is integer for many n, notably 163". If I compute Exp[pi * sqrt(163)], I get an extremely large number. Also if I interpret it as the base-pi root of n, I get incorrect results. The second 'error' I've found is in 'Other numeric curiosities', where it is stated that phi = -2 * sin(666), while this is incorrect. In fact, on the page of the golden ratio, it is mentioned that -phi = sin(666) + cos(6 * 6 * 6), which seems to work out. 30 August 2006

You are right but there is no error. Exp[pi * sqrt(163)] = e^40.10916999113251975535008362290414... = 262537412640768743.99999999999925... which is both a large number and close to being an integer.
Meanwhile sin(666°) = −sin(54°)= −sin(3π/10 radians) = −0.80901699... = −phi/2. At the same time cos(6*6*6°) = cos(216°) = −cos(36°) = −sin(54°) so the same value. --Henrygb 14:45, 22 November 2006 (UTC)

[edit] Spelling of 40

40 is the only number whose (English name) letters are ordered alphabetically.

This has nothing to do with mathematics - not even marginally - and everything to do with language (but only one particular language). It is in interesting fact to bring out at trivia sessions, but it is certainly not a "coincidence" (mathematical or any other kind), and I don't see how it has any place in this article. JackofOz 01:13, 25 July 2006 (UTC)

Agree --Alex ^talk 14:01, 25 July 2006 (UTC)

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