1.Is there a integer x(express in bits...), so that.
md5(x) = x
2.If there is, how many are there.
3.If there aren't any, what's the least amount of md5 need to be applied to any x resulting itself.

There are 2^128 possible x to consider. Yes, Beyond our computation power

Suppose that md5(x) is a function that randomly maps an x with another non-negative integer that's smaller than 2^128(not necessarily one to one). There are some expected values.

Expected number of x with that property out of all possible x with above assumption.
I would think it as pick x, then randomly pick a number from 0 to 2^128-1 for f(x). Each pick are mutually exclusive. What's the chance x = f(x)
There are 2^256 combination of (x, f(x)), 2^128 of them are x = f(x)
Then we have...... 1 out of 2^128.
Doesn't give us too much hope... haha... This problem is not likely to be answered brutal force unless I'm really lucky or md5 is designed with like 10^20 md5(x) = x.

I'm searching for an analytical method to do No.1
2 and 3 are likely only brutal forceable. While there isn't enough computational power to do it.

This problem can be generalized into any hash function that's complex(beyond normal humans' analytical thinking) and large(beyond any computer's computational power) to show how limited our knowledge are.

My site have been gaining people with strange fetishes.

I remember once heard that a spy in some scifi story stored the entire library of data from the military with a needle. He marked a point on the needle, which divide the needle into 2 parts. The length of each part became the numerator and denominator of an fraction, and the binary expansion of the fraction is the compressed data of the library.

How innovative. Or is it? Fit a large amount of data by converting it into a fraction and store it in a needle. How convenient. Why isn't it happening in real life?

One must note that currently, we can only manipulate the world discretely. We can't continuously divide the needle into infinite small parts. Assuming the smallest object can be precisely altered is an atom.
Let's make some assumptions about an ordinary needle. Suppose it have 4 cm long, and the diameter of an atom is 0.1nm and all very tightly packed. Then there are 40000 0000 atoms, 4*10^8. Call this number k.

WLOG, the first part of the needle has x atoms, the 2nd part has y atoms. It's ok to assume x != y. Why put the data on a needle when the number "1" is all he have to remember.

The binary digit expansion of x/y is the compressed data. The expansion digit is almost always infinite, but only the digits before the first repeat is useful.

What is the largest possible binary expansion for this?
Less than k.

So, the entire library can be stored in less than 50MB of data? Please...
And even if that IS the case, what's the possibility of that? there are 2^k combination of k bits. k/2^k chance. It's more likely to win the lottery and get hit by lightning... 3 times... in 20 seconds... than that the library actually can be stored with such an needle.

I don't know what sci-fi thing it is from. I wish to see the author and tell him how this is unpossible(p(x) = impossible ->p(x) = ε. p(x) = unpossible -> p(x) = 0)

Data From Salary.com

Y axis is amount of money in dollars
X axis is the percentile, only plot 3 points, 25th, 75th and 90th.
A line with the lowest starting point in each color means Instructor of [Insert Major].
2nd lowest is Assistant professor of [Insert Major].
3rd lowest is Associate professor of [Insert Major].
then it's Professor of [Insert Major].
Last one is Dean of [Insert Major].

From what I can see, clearly, the safe major to chose is computer science. Pay is good even for a CS instructor(around the same pay as a associate professor in philosophy, damn!

Philosophy is the most unsafe. Unless one can become a professor and manage to get into the very top, it's better to select the other 2.

Math is good. Money increase is steady compare to title.
Math professors can earn more than dean of mathematics.

Dean of medicine totally owns everyone else in colleges. The 25th percentile is already outside this graph.(almost 300K)

Update: It is, just not featured on multiplication algorithm page.
I just found it out today.
Someone should read it and understand it and wikipeida it. [That person is certainly, not me]
Faster Integer Mulitplication

We got No.2 St. Joseph's annual high school programming competition. No.1 get $500 for each teammate. OMG I wish I can get that money to cover my credit card debt.
No.1 and us are the only team to complete all problems. Had some really nice food.
MIT t-shirt and the PWNING NOOBZ headband.

Prize for been the lowest high scorer in math league. Epic fail -.-
Extreme Sports t-shirt. Math is an extreme sport.

Award for Journalism from Long Island Press High School Journalism Awards. Award for Format Buster... wonder what it is...
Evil grin. Have nothing to do with my "evilplanDoNotOpen" folder in my flash drive... or does it?...

So it seems, I'm more of a computer science person.