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Welcome 2014!

As we're on the verge of bidding farewell to \(2013\) and welcoming \(2014\) into our lives, I thought it was appropriate to do a post on the number \(2014\). So I googled and searched Wikipedia for anything relevant. But I found absolutely nothing. \(2014\)'s not a prime number, not a Bell number, not a Catalan number, not a Fibonacci number, not a factorial, not a perfect number, not a triangular number; nothing! Some numbers close to \(2014\) have some really interesting properties [for example, \(2011\) is a sexy prime number and \(2015\) is a Lucas-Carmichael number]. It seemed as though \(2014\) was the dullest number in the world.

So, I decided to take matters into my own hands. I started doing research on more subtle stuff on the internet and found out that \[2014=2013+1\] and \[2014\equiv\lfloor(\sum_{k=0}^{\infty}\frac{(2\pi)^{2k}}{(2k)!}(-1)^{2k}) \times \pi\times 10^{3136}\rfloor\pmod {10^4}\]. The second relation is due to the fact that the string '\(2014\)' occurs at position \(3331\) counting from the first digit after the decimal point in the decimal expansion of \(\pi\). In case you're wondering where the weird-looking infinite sum came from, it's just equal to \(1\). I put it there to make myself look smart :)

It's not surprising that the string '\(2014\)' appears in the decimal expansion of \(\pi\). It's likely that all finite sequences of numbers appear in \(\pi\)'s digits. I encourage you to read the post I linked right now and also look at the comments as well. There is a little chance that the third comment is by the great Grigori Perelman!

As I was saying, we have the year \(2014\) ahead of us. And even though I haven't been able to find anything really cool about the number \(2014\), I hope you've enjoyed reading this. Let us look forward to all the great things about to happen next year. For example, there's going to be an annular solar eclipse on April \(29\), \(2014\). And that's just one of the many things to come!

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Welcome \(2014\) and Happy New Year!

Note by Mursalin Habib
3 years, 2 months ago

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\(2014 = 13^3 - 13^2 - 13^1 - 13^0 \).

Not exactly a property, but it is a somewhat surprising fact. Calvin Lin Staff · 3 years, 2 months ago

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Most interesting thing about 2014 is

Date/Month/Year

4/4/2014 is Friday

6/6/2014 is Friday

8/8/2014 is Friday

10/10/2014 is Friday

12/12/2014 is Friday

Calendar of 2014 is same as that of 1947 and 1997 means 2014=1997=1947 Divyansh Singhal · 3 years, 2 months ago

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Perhaps the special thing about 2014 is that,while other numbers have something special about them,2014 doesn't! Rahul Saha · 3 years, 2 months ago

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@Rahul Saha But it's still an interesting number, because if it was an uninteresting number, then that would be an interesting quality about it! ;) Trevor B. · 3 years, 2 months ago

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Happy New Year everyone! :D Thank you for the great year 2013! I have learnt so much here on Brilliant - abt Math competitions stuff and also MORE on the beauty of Math (like the Pokemon graphs, Golden Ratio, mystery fractions, etc.) All the best to all for the next year, be it gd results in competition, exams, or anything you wish to accomplish! Most importantly, gd health and happiness! :D HAPPY NEW YEAR 2014!!! Happy Melodies · 3 years, 2 months ago

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What an amazing post! The fact that \(2014=2013+1\) blew my mind though, who would have thought! Ivan Sekovanić · 3 years, 2 months ago

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@Ivan Sekovanić Exactly! No other number has that property! \(2014\) is the only number that is equal to \(2013+1\) :)

Thanks by the way. Mursalin Habib · 3 years, 2 months ago

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@Mursalin Habib Not quite. 2013+1=2020 in base 4. Bam. :D Sharky Kesa · 2 years, 11 months ago

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@Sharky Kesa [To avoid confusion, all numbers in this comment are expressed in base 10 (\(2\times 5\)) [including this one]].

I'm afraid you don't get the point. Bases are just a way to represent numbers. A number could have different representations in different bases. In other words, it can look different but that doesn't change its properties. The numbers that you wrote may look like 2013 and 2020 in base 10 but they're not. They're actually 135 and 136 respectively [remember that the whole comment is in base 10]. What I wrote was if you add 'one' with 'two thousand and thirteen', you're going to get "two thousand and fourteen". Now I can express these numbers anyway I want [Hell, I can even draw an apple for 2013, draw a bicycle for the operation of addition, draw a bunny for 1 and put \(\pi\) as 2014! The only problem there is it's going to be hard to communicate with other people [and it's going to take a lot of time to express equations.]]. So, bottom line, my statement was correct. You were just reading it in a different language and had a misunderstanding. Mursalin Habib · 2 years, 11 months ago

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@Mursalin Habib I know. I was just playing around. Sharky Kesa · 2 years, 11 months ago

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2014 is the second year after 1987 that is made up of different digits. The first year is 2013. Still not that special. Bruce Wayne · 3 years, 2 months ago

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Happy New Year 2014 to everyone and let's hope the new year brings with itself a lot of interesting problems on Brilliant. Soham Dibyachintan · 3 years, 2 months ago

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Happy new year!!.Please see-(http://www2.stetson.edu/~efriedma/numbers.html) This is very interesting thing about the different numbers. Divyansh Singhal · 3 years, 2 months ago

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Nice post! Happy new year to you too! I have already seen the links you posted above, related to pi. They were quite interesting the first time, so I went and checked them out again! :)

I am sure you have heard about Ramanujan's number, 1729, and the interesting story behind it. If not, see http://en.wikipedia.org/wiki/1729_(number)

Let's hope someone here on Brilliant can come up with something as amazing for 2014! Here's to a great year for the whole Brilliant community! :D Rohan Rao · 3 years, 2 months ago

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Amazing !!! Sriram Venkatesan · 2 years, 6 months ago

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@Sriram Venkatesan Thanks! Mursalin Habib · 2 years, 6 months ago

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@Mursalin Habib Excellent! Swapnil Das · 2 years ago

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