Basically, post the weirdest, most beautiful or just the coolest mathematical equation you have ever come across. Here is an example:

\[e^{ix} = \cos{x} + i \sin{x}\]

Only one entry per person is allowed and the winner is determined by the number of up votes.
**Any down votes will be added on as up votes**. Good luck, have fun and do math.

P.S.

Who understands the abbreviation? Why is it significant?

## Comments

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TopNewestThis number amazed me:

\[1741725= 1^7+7^7+4^7+1^7+7^7+2^7+5^7\] – Hasan Kassim · 2 years, 9 months ago

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Narcissistic Number. There're even more than that! – Samuraiwarm Tsunayoshi · 2 years, 9 months ago

Check outLog in to reply

– Hasan Kassim · 2 years, 9 months ago

Wow they are much!! Thanks for sharing that :)Log in to reply

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– Agnishom Chattopadhyay · 2 years, 9 months ago

Why is it fascinating?Log in to reply

\(Oa = Ob , OA=OB \)

\(\frac{AB}{ab} =\frac{OA}{Oa}\)

\(\frac{Perimeter of the outer polygon}{Perimeter of the inner polygon} = \frac{n.AB}{n.ab} = \frac{AB}{ab}=\frac{OA}{Oa}\)

\(\frac{Circumference of outer circle}{Circumference of inner circle} = \frac{OA}{Oa} = \frac{Radius of outercircle}{radius of inner circle}\)

Thus

\( \frac{Circumference of outer circle}{radius of outercircle} = \frac{Circumference of inner circle}{radiusof inner circle}\)

You know \(\frac{Circumferenceofanycircle}{Diameter} =\) the constant\( \pi\)

The approximate values were given as \( \pi = \frac{22}{7}\) , more accurate was \(\frac{355}{113}\)

You can see now why it is fascinating@Agnishom Chattopadhyay – Megh Choksi · 2 years, 9 months ago

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– Agnishom Chattopadhyay · 2 years, 9 months ago

Hahahaha. I was asking why Ramanujan's formula for pi is interesting to you. Isn't the one than comes from arctan more simple and beautiful?Log in to reply

Which one is that which comes from arctan that is more simple and beautiful? – Megh Choksi · 2 years, 9 months ago

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To get this, you should realise arctan(1)=pi/4 and the. Just substitute arctan(1) with its taylor series.

However, your formula is better because it has faster convergence) – Agnishom Chattopadhyay · 2 years, 9 months ago

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Pi, written as π, equals 3.1415926… and its digits goes on forever, without any repeating pattern. Numbers with this property are called irrational numbers. Many ancient mathematicians – including the famous Pythagoras – were horrified when they discovered that such bizarre and ‘impure’ numbers exist.

Today many mathematicians believe that Pi has an even more curious property: that it is a normal number. This would mean that the digits from 0 to 9 appear completely at random, as if nature had rolled a 10-sided dice, again and again, to find the next digit. It also means that if you think of any string of digits, like 123456789, it has to appear somewhere in the digits of Pi – but you might have to calculate millions of digits.

We could even convert an entire book, like the works of Shakespeare, into a very long string of digits (a = 01, b = 02, and so on). If Pi is normal, this string must also appear somewhere in its digits. But even if we used all computers on Earth to calculate more and more digits of Pi, we would probably have to look for longer than the age of the universe… The First few Digits of Pi

3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628

0348253421170679821480865132823066470938446095505822317253594081284811174502841027019

3852110555964462294895493038196442881097566593344612847564823378678316527120190914564

8566923460348610454326648213393607260249141273724587006606315588174881520920962829254

0917153643678925903600113305305488204665213841469519415116094330572703657595919530921

8611738193261179310511854807446237996274956735188575272489122793818301194912983367336

2440656643086021394946395224737190702179860943702770539217176293176752384674818467669

4051320005681271452635608277857713427577896091736371787214684409012249534301465495853

7105079227968925892354201995611212902196086403441815981362977477130996051870721134999

9998372978049951059731732816096318595024459455346908302642522308253344685035261931188

1710100031378387528865875332083814206171776691473035982534904287554687311595628638823

53

There are many different ways to calculate Pi, some of which use sequences or series of numbers. One example is the following series discovered by Gottfried Wilhelm Leibniz (1646 – 1716). As you add more and more terms, following the same pattern, the result will get closer to Pi:

\(π = \frac{4}{1} – \frac{4}{3} + \frac{4}{5} – \frac{4}{7} + \frac{4}{9} – \frac{4}{11} + …\)

Another sequence, published by Nilakantha Somayaji (1444 – 1544), is even better since it gets closer to Pi with fewer terms:

\(π = 3 + \frac{4}{2 × 3 × 4} – \frac{4}{4 × 5 × 6} + \frac{4}{6 × 7 × 8} – \frac{4}{8 × 9 × 10} + …\)

This formula was published by John Wallis in 1655:

\(π = 2 × \frac{2}{1} × \frac{2}{3} × \frac{4}{3} × \frac{4}{5} × \frac{6}{5} × \frac{6}{7} × \frac{8}{7} × \frac{8}{9} × …\)

Using powerful computers, Pi has been calculated up to 10 trillion digits (that’s a 1 with 13 zeros)! Because Pi is so easy to understand, yet important in many areas of mathematics, it enjoys an unusual popularity in our culture (unusual, at least, for areas of mathematics). There even is a Pi Day on 14 March or 22 July, since \(\frac{22}{7}\) is a close approximation to pi. @Agnishom Chattopadhyay – Megh Choksi · 2 years, 9 months ago

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– Agnishom Chattopadhyay · 2 years, 9 months ago

Hm, I remember wishing 'Happy Pi Day' to my beloved onesLog in to reply

– Kartik Sharma · 2 years, 9 months ago

Prove it! :P Does anyone has a proof of it?Log in to reply

I have a proof for my formula – Agnishom Chattopadhyay · 2 years, 9 months ago

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– Kartik Sharma · 2 years, 9 months ago

Ramanujan's formula!Log in to reply

– Agnishom Chattopadhyay · 2 years, 9 months ago

I have notLog in to reply

For \[n = 1,2,3,4,5,6\] is true that:

\[14^n+16^n+45^n+54^n+73^n+83^n = 3^n+ 5^n + 28^n + 34^n+ 65^n + 66^n + 84^n\] – Jordi Bosch · 2 years, 9 months ago

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– Kartik Sharma · 2 years, 9 months ago

Can you tell me why is this happening?Log in to reply

here to see more than that. – Samuraiwarm Tsunayoshi · 2 years, 9 months ago

Probably magic. You can checkLog in to reply

– Kartik Sharma · 2 years, 9 months ago

Nice! Thanks for sharing!Log in to reply

A number which rearrange itself upto multiplication by 6

142857 \(\times\) 1 = 142857

142857 \(\times\) 2 = 285714

142857 \(\times\) 3 = 428571

142857 \(\times\) 4 = 571428

142857 \(\times\) 5 = 714285

142857 \(\times\) 6 = 857142

Here is the interesting part

142857 \(\times\) 7 = 999999 – Krishna Sharma · 2 years, 9 months ago

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– Christopher Boo · 2 years, 9 months ago

\(\frac{1}{7}=0.142857...\)Log in to reply

– Sharky Kesa · 2 years, 9 months ago

Cyclic number.Log in to reply

\(\displaystyle e^{\pi \sqrt(163)} \approx 262537412640768743.99999999999925007 \approx 640320^3+744-0.00000000000075\)

Coincidence

What do you think?

Credits : Tumblr (& The Incredibles), Ramanujan, Wikipedia. – B.S.Bharath Sai Guhan · 2 years, 9 months ago

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– Christopher Boo · 2 years, 9 months ago

What's so special about this equation?Log in to reply

Is that a coincidence or what?!

P.S: You might wanna check out 163 and Ramanujan's constant. – B.S.Bharath Sai Guhan · 2 years, 9 months ago

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– Agnishom Chattopadhyay · 2 years, 9 months ago

What does that even mean?Log in to reply

P.S: You might want to check out Almost Integer for more such 'coincidences'. – B.S.Bharath Sai Guhan · 2 years, 9 months ago

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From http://math.stackexchange.com/questions/8814/funny-identities?rq=1

\[ \frac{\pi}{2} = \frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\ldots\\ \frac{\pi}{4} = 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\ldots\\ \frac{\pi^2}{6} = 1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\ldots\\ \frac{\pi^3}{32} = 1-\frac{1}{3^3}+\frac{1}{5^3}-\frac{1}{7^3}+\frac{1}{9^3}+\ldots\\ \frac{\pi^4}{90} = 1+\frac{1}{2^4}+\frac{1}{3^4}+\frac{1}{4^4}+\frac{1}{5^4}+\ldots\\ \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\ldots\\ \pi = \cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\ldots}}}}}\\ \] – Agnishom Chattopadhyay · 2 years, 9 months ago

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I loved this one as well.

\[\dfrac {d}{dx} (e^x) = e^x\] – Sharky Kesa · 2 years, 9 months ago

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Honestly, the equation that comes into my mind was

\[x^2-x-1=0\]

which is related to the golden ratio and the Fibonacci Sequence. – Christopher Boo · 2 years, 9 months ago

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Fermat's last theorem.. \(a^{n}+b^{n}=c^{n}\) for all a,b,c,n being integers gives real integral solutions only if n<=2 . The proof of the solution took 7 years of hard solitary work!!! – Ankit Chatterjee · 2 years, 9 months ago

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\(a^2 + b^2 = c^2\) (Pythagoras Theorem)

Edit-These positive integers \(a,b,c\) form the sides of a right triangle necessarily. – Anuj Shikarkhane · 2 years, 9 months agoLog in to reply

Inclusion-Exclusion: Either you like something, or you don't.

If you like this, then upvote it, or else, if you don't like it, then downvote it. – Satvik Golechha · 2 years, 9 months ago

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– Kartik Sharma · 2 years, 9 months ago

For the first time, there is not a single downvote(see!) in such an open discussion.Log in to reply

- You haven't read the note carefully. Despite me bolding the text, your despicable eyes have failed to read it -- – Krishna Ar · 2 years, 9 months agoLog in to reply

– Agnishom Chattopadhyay · 2 years, 9 months ago

They might be added but I still have downvoted comments I do not like.Log in to reply

– Krishna Ar · 2 years, 9 months ago

So it was you who did that -_-Log in to reply

– Agnishom Chattopadhyay · 2 years, 9 months ago

Yes, I downvoted -1/12 because I can't stand these Kaboobly Doo ideas.Log in to reply

– Curtis Clement · 2 years, 6 months ago

It's not a; kaboobly; idea - there is proof of the outcome. It just seems counter-intuitive because it would seem to equal infinity. However, that is only because you are imagining what happens if you stop it at the 10th terms, 100th term, 1000th term etc.Log in to reply

– Krishna Ar · 2 years, 9 months ago

What Irony -_- (says master of KD)Log in to reply

See this – Julian Poon · 2 years, 9 months ago

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It is similar to saying " 11 + 2 = 1 " (where the arithmetic is performed on the clock)

It is similar to saying "1 + 1 = 0 ". (Where the arithmetic is performed in the field of order 2)

Neither of these would be "acceptable", and are only technically right by omission. I value clarity of expression much more than showing off. – Calvin Lin Staff · 2 years, 9 months ago

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youwho did that. – Sharky Kesa · 2 years, 9 months agoLog in to reply

– Satvik Golechha · 2 years, 9 months ago

So who was it who did WHAT?!Log in to reply

– Sharky Kesa · 2 years, 9 months ago

Bolded that sentence.Log in to reply

Iedited this comment of yours too :P – Krishna Ar · 2 years, 9 months agoLog in to reply

– Sharky Kesa · 2 years, 9 months ago

Yeah there is. Check out Samuraiwarm's comment below. 1 downvote.Log in to reply

– Krishna Ar · 2 years, 9 months ago

I neither like it nor dislike it :3Log in to reply

– Satvik Golechha · 2 years, 9 months ago

Expected from you. You can still upvote it.Log in to reply

@Sharky Kesa I just trolled your meow... – Satvik Golechha · 2 years, 9 months ago

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@Sharky Kesa I just trolled your (meow)

I just trolled your what? – Sharky Kesa · 2 years, 9 months ago

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– Satvik Golechha · 2 years, 9 months ago

Depends.Log in to reply

Read this slowly:

MATHISNOWHERE

What does it say? – Sharky Kesa · 2 years, 9 months ago

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– Anuj Shikarkhane · 2 years, 9 months ago

MATH IS NOWHERE OR MATH IS NOW HERE. :DLog in to reply

– Sharky Kesa · 2 years, 9 months ago

Which one do you prefer?Log in to reply

– Anuj Shikarkhane · 2 years, 9 months ago

I prefer "MATH IS NOW HERE"Log in to reply

read like "matt, his now - hurry" – Justin Wong · 2 years, 9 months ago

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– Julian Poon · 2 years, 9 months ago

I've got to note that one...Log in to reply

– Satvik Golechha · 2 years, 9 months ago

Depends.Log in to reply

– Sharky Kesa · 2 years, 9 months ago

Which one do you prefer?Log in to reply

– Satvik Golechha · 2 years, 9 months ago

Depends.Log in to reply

– Sharky Kesa · 2 years, 9 months ago

How so?Log in to reply

– Satvik Golechha · 2 years, 9 months ago

That too depends.Log in to reply

– Sharky Kesa · 2 years, 9 months ago

List why it depends.Log in to reply

– Satvik Golechha · 2 years, 9 months ago

It actually doesn't depend. I was just replying "Depends" for fun.Log in to reply

– Sharky Kesa · 2 years, 9 months ago

So then, answer my question.Log in to reply

– Satvik Golechha · 2 years, 9 months ago

Which question? You've asked more than one.Log in to reply

– Sharky Kesa · 2 years, 9 months ago

All of them.Log in to reply

– Agnishom Chattopadhyay · 2 years, 9 months ago

That is a very good idea,Log in to reply

\[\sin{(2°\times{10}^{-n})}-\sin{(2°\times{10}^{-(n+1)})}\approx\pi\times{10}^{-(n+2)}\]

Try to derive it and you'll see the magic. – Julian Poon · 2 years, 9 months ago

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– Calvin Lin Staff · 2 years, 9 months ago

Oh, nice. LOLLog in to reply

\({ G }_{ \mu v }=8\pi G\left( { T }_{ \mu v }+{ \rho }_{ \Lambda }{ g}_{ \mu v } \right) \)

Einstein field equation– Shivamani Patil · 2 years, 9 months agoLog in to reply

1+(e^(i*pi))=0

five fundamental numbers in one equation... almost all you guys are already familiar with this – Rohan Asif · 2 years, 9 months ago

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Taylor Series -

\(f(x) = \sum _{ n\quad =\quad }^{ \infty }{ \frac { { f }^{ (n) }(0) }{ n! } { x }^{ n } } \)

I have one more(if it is allowed):

https://qph.is.quoracdn.net/main-qimg-7beca9cf6c835bdfeabac3541a778973?convert

towebp=trueAnd one more:

http://en.wikipedia.org/wiki/Tupper%27s

self-referentialformula*The last 2 are just for sharing – Kartik Sharma · 2 years, 9 months ago

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\[\zeta(-1) = 1+2+3+\dots = -\frac{1}{12}\]

where \(\zeta(s)\) is the Riemann zeta function. – Samuraiwarm Tsunayoshi · 2 years, 9 months ago

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Otherwise, it is similar to saying " 1 + 1 is equal to 10 "

\( \vdots \)

(where I am working in binary) – Calvin Lin Staff · 2 years, 9 months ago

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– Curtis Clement · 2 years, 8 months ago

The Riemann Zeta function also links with quantum mechanics and is said to encode a 'formula' for the distribution of primes. Whether it does is the million dollar question!Log in to reply

– Julian Poon · 2 years, 9 months ago

wait waa?Log in to reply

it is...........................

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1 = 1 – Math Man · 2 years, 9 months ago

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@math man – Anuj Shikarkhane · 2 years, 9 months ago

what is so great in that?Log in to reply

– Math Man · 2 years, 9 months ago

idk lolLog in to reply

– Anuj Shikarkhane · 2 years, 9 months ago

:DLog in to reply

This is not for competition. However this is what I remember since I was of teen age.\(\color{blue}{152207 * 73= 11111111.(Eight~ '1's)\\~ It~ is ~ clear~ that ~if ~we~use~N*73, N=1,2,...9,~~\\we~ get~ EIGHT~ Ns.~\\152207~is ~divisible ~by ~11. ~(11|152207)} \) – Niranjan Khanderia · 2 years, 6 months ago

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\(V-E+F=2\) – Danny Kills · 2 years, 9 months ago

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– Parth Lohomi · 2 years, 9 months ago

Euler's formulaeLog in to reply

– Hasan Kassim · 2 years, 9 months ago

And what is that about?Log in to reply

substitute x=pi. then eqution becomes e^(i pi)+1=0. this relation connects 5 most important numbers of mathematics, which makes this most beautiful relation. :) – Thushar Mn · 2 years, 6 months ago

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\[\Im(i^{i})=0\] – Pranjal Jain · 2 years, 8 months ago

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You all have really good entries. I don't know I only have one equation that may seem not as good as yours. Still have a look. e^{i(pi)}= -1 This equation is derived from De Moivre's Theorem(Sorry Sharky Kesa) But I loved it more than the original theorem since it is of the form An irrational number raised to the product of an imaginary and irrational number which gives us a real number. P S I don't know how to write 'pi' in Greek. – Usama Khidir · 2 years, 8 months ago

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Haha...The abbreviation is Meow :) – Tan Li Xuan · 2 years, 9 months ago

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The sequence

1-1+1-1+1-1...an Oscillatory series equals 1/2 even though all are integers . – Keshav Tiwari · 2 years, 9 months agoLog in to reply

– Ninad Akolekar · 2 years, 9 months ago

Interesting! But can you explain why that happens?Log in to reply

– Keshav Tiwari · 2 years, 8 months ago

Refer to this note https://brilliant.org/discussions/thread/interesting-sums/ . Hope it helps ! :)Log in to reply

– Ninad Akolekar · 2 years, 8 months ago

Thanks!Log in to reply