Best way to express 2014

Looking $2014$ up in the OEIS turns up:

$$2014=13^3-13^2-13^1-13^0$$

In general, looking a number up in the OEIS is probably a reasonable way to turn up pleasing identities.

Give them the following :

$$(-2+0+1+4)^{(2+0+1+4)}-(2+0+1+4)^{(-2+0+1+4)}+(2+0-1+4)^{(-2+0+1+4)}+(2+0-1+4)\cdot(-2+0+1+4)^2=?$$

and tell them to compute the result.

All you can see is only 2014 with some sign changed and of course the result is simply
$$3^7-7^3+5^3+5\cdot3^2=2014$$
It will look better on a board.

With all due credit to this base 13 answer on codegolf.SE:

$$2014=BBC_{13}$$

Or just playing with my calculator, I like the look of

$$2014=5^5-1111$$

I hope you will enjoy the following spoof :

$\qquad\quad$ I remember once going to see him for the Holidays, and remarked that the number of the upcoming year seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. "No," he replied, "it is a very interesting number; it is the smallest number which can be expressed as the product of three distinct primes, which are congruent modulo $17$." $:)$

What it lacks in brevity, it makes up for in nerdiness:

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS$0$

Here are some cryptic ones: From the Gaussian integral we have $$2014 = \frac{4028}{\sqrt{\pi}}\int_{0}^{\infty}e^{-x^2}dx$$ and from the Basel Problem we have: $$2014 = \frac{12084}{\pi^2}\sum_{i=1}^{\infty}{\frac{1}{i^2}}$$ Here are some that (arguably) has deep meanings and roots: $$ 2014 = 2\cdot19\cdot53$$ $$2014 = 2^{11} - 34$$ For some trigonometry we have: $$2014 = \frac{4}{\cos^3{\frac{\pi}{9}}\cdot\cos^3{\frac{2\pi}{9}}\cdot\cos^3{\frac{4\pi}{9}}} - 34$$ It depends on perception, really. There are probably arguably infinitely many ways to write $2014$ in a "short, snappy, cool, and nerdy way".

How about

$$3\cdot6!-5!-4!-2!$$

or, if you like

$$(6!-5!)+(6!-4!)+(6!-2!)$$

Alternatively:

$$6!2!+4!4!-2!0!$$

(Self answering question) Find the integral part of the unique real root of the equation $$\log_2 x+\log_{20}x+\log_{201}x+\log_{2014}x -2-0-14+\frac{1}{20+\frac{1}{14}} = 0$$

Maybe do:

$$2014 = \sum_{k=0}^{11}\binom{11}{k}-34$$

$$ 2014 = 2^{2\times2^2 \times (2\times2^2-2)}-2\times (2\times2)^2-2 $$

Equations like this can be made for any number, not just 2014.

Binary: 11111011110 Hexadecimal: 7DE

Image the students perplexing expression when they see: 1101/11/111111011110 or C/3/7DE Tell them to write this date in this form on there notes. Guaranteed they will show it to their friends or family.

Wow I can't believe I haven't thought of doing this with my students. As a rookie high school math teacher I am always looking for new 'hooks' with my students. Great idea and thank you!

Lesson planning using stack exchange? Who knew..

If you're looking for "fascinating features" and you found only " too longwinded" formulae then you have for free the fascinating feature that 2014 is the first number for which you were not able to find a formula that is not "too longwinded"