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Pretty sure you mean not equal to 1 either. 0 gives a point and 1 gives a line, (which can be interpreted as a circle of infinite radius).

bump.

pressure.

Your wording is a little confusing but if I interpreted your question correctly then the answer is yes: One could use a technique similar to induction to prove a proposition about all integers by showing that S(0) is true, and then showing that: S(n+1) is true whenever S(n) is true. S(n-1) is...

$For an arbitrary sequence: $u=(u_1,u_2,\ldots)$ of complex numbers, define its sequence of differences by: $u'_n:=u_{n+1}-u_n$.\\ Let $u_n^{(k)}$ denote the $n$-th term of the sequence obtained by taking the sequence of differences of the sequence $u$ $k$-times recursively. (So for example...

Ditto!

http://www.youtube.com/watch?v=3ZOb2yE4QP0

I beg to differ, there is no argument about it! All the other rappers ain't got nothin' on my boy Bangs.

Haha all his stuff is gold. Lyrical genius.

Stumbled across a pretty funny mathematical drinking game / problem in graph theory posted by someone on Terence Tao's blog: Say you have a group of n people. The player who is "it" says an integer k strictly between 1 and n+1 and at the same time everyone including this player points at...

http://www.youtube.com/watch?v=IfHXDZsVTYA

Didn't expect such low marks for Q6...

\overline{2}\neq 1/2.

For some people it does, and who am I to say that it is "wrong" to pursue such things? I am more bothered by people who DON'T enjoy this lifestyle yet throw away dreams for the sake of conformity, than those who truly do enjoy the rat race. (Pretty sure your views are quite similar to mine on...

Equivalently: How many ways can we put the n students into two rooms? 2^n. (there are two choices of room for each student.) How many of these situations involve an empty room? 2. (the situations where they are all in room A and where they are all in room B.) So the answer is: 2^n-2.

On a large scale competitiveness is indeed what drives our capitalist world and makes the wheels of society go round, but I think it is very much up to the individual to discover what is truly important to him/herself and what role he wants to play in this world. There is nothing wrong with...

Coffee is my lifeblood. Have used energy drinks many times and dex a couple of times but I think that being healthy trumps pretty much any substance when it comes to getting work done at a consistent rate.

Here are two: Fermat's Last Theorem N is a Number: A Portrait of Paul Erdos.

My bad, apologies for the rushed and incorrect work.

There are no non-integer rational solutions by the rational root theorem. In fact I claim that the only rational solutions are: (0,0), (0,1), (1,0), (1,1). Outline of proof: -Establish the result for non-negative integers using that x^3 >= x^2 for non-negative integers with equality iff x=0 or...