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!

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Welcome 2014!

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