Best Theorems!

ITT best theorems you know so far

Picard's Theorem is pretty nifty.

Any entire complex function that omits two values in a constant.

Stokes Theorem.  The generalization of many useful theorems.

The Chinese Remainder Theorem.  Oldy but a goody.

Pythagorean Theorem / Law of Cosines

Halting Problem.

One of my favorites would have to be Euclid's proof of the infinitude of primes.  Simple, straightforward, and elegant.

The proof that .999999... equals to 1

>>6
Indeed, I find that pretty nifty too.

It's also true that for any integer k, there exists a sequence of k numbers, none of which are prime.

Proof:
The integers between [(k+1)! + 2] and [(k+1)! + k+1] form such a sequence

sum of consecutive squares from 1^2 to n^2 equals the cube of the sum of consecutive numbers from 1 to n
I found it by accident and thought, hey neat

whoops I mean sum of consecutive cubes from 1^3 to n^3 equals the square of the sum of consecutive numbers

I like completing the square to generate the quadratic equation. To me it represents the essence of maths. The quadratic equation is a whole new level to the term "a priori" and illustrates the importance and elegance of mathematics and of the fundamentals of the natural world. It is not drawn from observation of the world like counting or just an elaboration of already existing concepts like integers, it is an entirely new concept derived from the core a posteriori mathematical concepts.

There are other theorems with the same qualities, but completing the square was the first one I recognized to be important and is a familiar example.

>>1
Captain Picard
CP
PEDO

>>9/10
Nicomachus's Theorem

Hairy Ball Theorem

"One cannot comb the hairs on a ball in a smooth manner."

>>14
its actually not very insightful nor complicated.

Nigger Theorem: if you see a nigger riding anything that's not a skateboard, it is stolen.

Taylor Series are pretty cool, and its implication that any analytical segment of a function can be represented by an infinite-degree polynomial

>>8
>Proof:
>The integers between [(k+1)! + 2] and [(k+1)! + k+1] form such a sequence

That doesn't quite prove what you said...

The sequence I constructed consists only of composite numbers, and there are k of them.  Hence it is proven.

>>19
Ah, I see what you are trying to say.  The sequence of numbers of the form [(k+1)! + k+1] where k runs through the positive integers.

Cool. :)

>>19
Easier might be just n! (n=3,4,5,6,7,...), and certainly it's not hard to think of other examples.

>>17
that only works properly for complex functions.

there are analytic real functions whos taylor series doesn't represent the function.

du u (10)∫(13) 2x dx   ???

>>22
Uh what? The definition of an analytic function on an open subset D of the reals is precisely that it can be expressed as a power series on D. The "definition" you're probably thinking of is that Analytic on D <=> C1 on D, which in actuality is just a consequence of the real definition of analytic in the complex plane.

>>21
>>20
I think >>19 and >>9 mean a sequence of _consecutive_ integers, as otherwise the problem is trivial.

>>24
oops; thanks for the correction.  When I said sequence, I automatically assumed consecutive, which is not a good assumption.